GCSE Maths: The equation y = mx + c (Higher)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
The equation y = mx + c
Every straight line (that is not vertical) has an equation of the form y = mx + c. The two letters m and c are numbers that describe the line:
- m is the gradient: the number multiplying x. It tells you how steep the line is and which way it slopes.
- c is the y-intercept: the number on its own. It is the value of y where the line crosses the y-axis, because on the y-axis x = 0 and so y = m × 0 + c = c.
For example, y = 3x + 2 has gradient 3 and crosses the y-axis at (0, 2). The line y = −x + 5 has gradient −1 (a minus sign on its own means −1) and y-intercept 5. The line y = 4x has no c showing, so c = 0 and it passes through the origin. Watch the order: in y = 5 − 3x the gradient is −3 and the intercept is 5, because each sign belongs to the number after it.
Sketching a line using m and c
You can sketch a line without a table of values. Step 1: mark the y-intercept (0, c). Step 2: use the gradient as “across then up (or down)” to find another point: for a gradient of 2 = 2/1 go 1 across and 2 up; for −3/4 go 4 across and 3 down. Step 3: join the points with a ruler.
Lines that are not written as y = …
If the equation is not in the form y = mx + c, rearrange it first so that y is on its own. For example 2y = 6x + 4: divide every term by 2 to get y = 3x + 2. And y − 2x = 3: add 2x to both sides to get y = 2x + 3. You met rearranging in Chapters 8 and 9.
Finding the axis crossing points
A line crosses the y-axis where x = 0 and the x-axis where y = 0. For equations such as 2x + 3y = 12 this is a quick way to sketch: put x = 0 to get 3y = 12 so y = 4, giving (0, 4); put y = 0 to get 2x = 12 so x = 6, giving (6, 0). Join the two points.
Higher: you will be asked to rearrange equations such as 3x + 4y = 12 into the form y = mx + c to find the gradient and intercept. The steps are: subtract 3x from both sides to get 4y = −3x + 12, then divide every term by 4 to get y = −34x + 3. The gradient is −34 and the intercept is 3.
Key ideas
- y = mx + c: m = gradient (number multiplying x), c = y-intercept (the point (0, c)).
- Keep the sign with the number: in y = 5 − 3x, m = −3 and c = 5.
- Sketch: plot (0, c), then use the gradient (across, then up or down), then join with a ruler.
- If y is not on its own, rearrange first. Divide every term.
- The y-axis crossing: x = 0. The x-axis crossing: y = 0.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = −2x + 7
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −2
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 7, so the line crosses the y-axis at (0, 7)
Check. Put x = 0 into the equation: y = 7, which is the y-intercept. Put x = 1: y = 5, which is −2 more than 7, so the gradient is −2. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Write the equation in the form y = mx + c. Put the x term first and the plain number last, keeping the sign in front of each term.y = 5 − 3x = y = −3x + 5
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −3
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 5, so the line crosses the y-axis at (0, 5)
Check. Put x = 0 into the equation: y = 5, which is the y-intercept. Put x = 1: y = 2, which is −3 more than 5, so the gradient is −3. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Start at the y-intercept, then use the gradient (rise ÷ run) to find more points. No table of values is needed.
- Find the y-intercept and mark it. The number on its own is c. The line crosses the y-axis at (0, c), so this is the first point.c = −2: mark (0, −2)
- Use the gradient to find a second point. The gradient is 12, which means: go 2 across and 1 up (rise ÷ run = 1/2).from (0, −2) go 2 right and 1 up to (2, −1)
- Go the other way to get a third point as a check. Going left by the same amount means doing the opposite of the rise too.from (0, −2) go 2 left and 1 down to (−2, −3)
- Draw a straight line through the points with a ruler. Extend the line across the grid. Label it with its equation.
Check. Pick the point with x = 2: y = 12 × 2 − 2 = −1, which is on the sketch. ✓
Watch out. For a negative gradient go down as you go across (or up if you go left).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Divide every term by the number in front of y to get the form y = mx + c.
- The equation has a number multiplying y. We want y on its own (coefficient 1), so divide both sides by that number.Start: 2y = 6x + 4
- Divide every term by 2. Every term on the right must be divided too, not just the first.y = 6x ÷ 2 + 4 ÷ 2
y = 3x + 2 - Read off the gradient and the y-intercept. Compare with y = mx + c.m = 3, c = 2
Check. Test x = 2: the original gives 2y = 16, so y = 8. The new equation gives y = 3 × 2 + 2 = 8. ✓
Watch out. Divide every term by the number in front of y, including the plain number.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: 3x + 4y = 12
Subtract 3x from both sides: 4y = −3x + 12 - Divide every term by the number in front of y. y must have a coefficient of 1. Divide every term (both of them) by 4.Divide every term on both sides by 4: y = −34x + 3
- Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = −34, c = 3
Check. Test x = 3. Original: 3 × 3 + 4 × y = 12 gives y = 0.75. New form: y = (−34) × 3 + 3 = 0.75. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The x-intercept is where the line crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
Method. Find where the line crosses each axis by putting x = 0 and then y = 0. Join the two points.
- Find where the line crosses the y-axis: put x = 0. Every point on the y-axis has x = 0. The x term disappears.3y = 12 → y = 12 ÷ 3 = 4
The line crosses the y-axis at (0, 4). - Find where the line crosses the x-axis: put y = 0. Every point on the x-axis has y = 0. The y term disappears.2x = 12 → x = 12 ÷ 2 = 6
The line crosses the x-axis at (6, 0). - Plot the two points and join them with a ruler. Two points are enough for a straight line. Extend the line beyond them.
Check. Test a third point, such as x = 8: y = (12 − 2 × 8) ÷ 3 = −43, and the point is on the sketch. ✓
Watch out. For the y-intercept the x value is 0; for the x-intercept the y value is 0. Do not mix them up.
Watch out
Signs and order. The sign belongs to the term after it. In y = 5 − 3x, the gradient is −3 (not 3), and the y-intercept is 5 (not −3). And when you rearrange, divide every term: 2y = 6x + 4 gives y = 3x + 2, not y = 3x + 4.Exercise 12F Gradient, intercept and sketching lines
Non-calculatorKey words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = 2x + 3
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = 2
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 3, so the line crosses the y-axis at (0, 3)
Check. Put x = 0 into the equation: y = 3, which is the y-intercept. Put x = 1: y = 5, which is 2 more than 3, so the gradient is 2. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = 5x − 1
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = 5
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = −1, so the line crosses the y-axis at (0, −1)
Check. Put x = 0 into the equation: y = −1, which is the y-intercept. Put x = 1: y = 4, which is 5 more than −1, so the gradient is 5. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = −x + 4
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −1
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 4, so the line crosses the y-axis at (0, 4)
Check. Put x = 0 into the equation: y = 4, which is the y-intercept. Put x = 1: y = 3, which is −1 more than 4, so the gradient is −1. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = 12x − 2
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = 12
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = −2, so the line crosses the y-axis at (0, −2)
Check. Put x = 0 into the equation: y = −2, which is the y-intercept. Put x = 1: y = −1.5, which is 0.5 more than −2, so the gradient is 0.5. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = 3x
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = 3
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 0, so the line crosses the y-axis at (0, 0)
Check. Put x = 0 into the equation: y = 0, which is the y-intercept. Put x = 1: y = 3, which is 3 more than 0, so the gradient is 3. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Compare with y = mx + c. The equation is already in the form y = mx + c, so we can read it off.y = −4x + 6
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −4
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 6, so the line crosses the y-axis at (0, 6)
Check. Put x = 0 into the equation: y = 6, which is the y-intercept. Put x = 1: y = 2, which is −4 more than 6, so the gradient is −4. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Use y = mx + c: the gradient replaces m and the y-intercept replaces c.
- Write the general equation of a line. m is the gradient and c is the y-intercept.y = mx + c
- Put in the gradient. Replace m by 4.y = 4x + c
- Put in the y-intercept. The line crosses the y-axis at (0, −3), so c = −3. Remember to use the sign.y = 4x − 3
Check. When x = 0, y = −3. When x goes up by 1, y changes by 4. ✓
Watch out. A negative c is subtracted: c = −3 gives y = … − 3.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Use y = mx + c: the gradient replaces m and the y-intercept replaces c.
- Write the general equation of a line. m is the gradient and c is the y-intercept.y = mx + c
- Put in the gradient. Replace m by −2.y = −2x + c
- Put in the y-intercept. The line crosses the y-axis at (0, 5), so c = 5. Remember to use the sign.y = −2x + 5
Check. When x = 0, y = 5. When x goes up by 1, y changes by −2. ✓
Watch out. A negative c is subtracted: c = −3 gives y = … − 3.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Use y = mx + c: the gradient replaces m and the y-intercept replaces c.
- Write the general equation of a line. m is the gradient and c is the y-intercept.y = mx + c
- Put in the gradient. Replace m by 1.y = x + c
- Put in the y-intercept. The line crosses the y-axis at (0, 0), so c = 0. Remember to use the sign.y = x
Check. When x = 0, y = 0. When x goes up by 1, y changes by 1. ✓
Watch out. A negative c is subtracted: c = −3 gives y = … − 3.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Use y = mx + c: the gradient replaces m and the y-intercept replaces c.
- Write the general equation of a line. m is the gradient and c is the y-intercept.y = mx + c
- Put in the gradient. Replace m by 13.y = 13x + c
- Put in the y-intercept. The line crosses the y-axis at (0, 2), so c = 2. Remember to use the sign.y = 13x + 2
Check. When x = 0, y = 2. When x goes up by 1, y changes by 13. ✓
Watch out. A negative c is subtracted: c = −3 gives y = … − 3.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Write the equation in the form y = mx + c. Put the x term first and the plain number last, keeping the sign in front of each term.y = 5 − 3x = y = −3x + 5
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −3
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 5, so the line crosses the y-axis at (0, 5)
Check. Put x = 0 into the equation: y = 5, which is the y-intercept. Put x = 1: y = 2, which is −3 more than 5, so the gradient is −3. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Write the equation in the form y = mx + c. Put the x term first and the plain number last, keeping the sign in front of each term.y = −1 + 2x = y = 2x − 1
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = 2
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = −1, so the line crosses the y-axis at (0, −1)
Check. Put x = 0 into the equation: y = −1, which is the y-intercept. Put x = 1: y = 1, which is 2 more than −1, so the gradient is 2. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Write the equation as y = mx + c. Then m is the gradient and c is the y-intercept.
- Write the equation in the form y = mx + c. Put the x term first and the plain number last, keeping the sign in front of each term.y = 7 − x = y = −x + 7
- The gradient m is the number multiplying x. Include its sign. If there is no number, x means 1x and m = 1; if there is only a minus sign, m = −1.m = −1
- The y-intercept c is the number on its own. It is where the line crosses the y-axis. If there is no number, c = 0 and the line goes through the origin.c = 7, so the line crosses the y-axis at (0, 7)
Check. Put x = 0 into the equation: y = 7, which is the y-intercept. Put x = 1: y = 6, which is −1 more than 7, so the gradient is −1. ✓
Watch out. Keep the sign with the number. In y = 5 − 3x the gradient is −3 (not 3) and the intercept is 5 (not −3).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Match using the y-intercept first, then the sign of the gradient.
- Find the y-intercept of the equation. The number on its own, c, is where the line crosses the y-axis.y = −x + 3: c = 3, so the line passes through (0, 3)
- Check the sign of the gradient. A positive m means the line goes up from left to right; a negative m means it goes down.m = −1 is negative: the line goes down
- Pick the line on the graph that matches both. Check the crossing of the y-axis and the direction.Line B crosses the y-axis at 3 and goes down.
Check. No other line crosses the y-axis at 3: the others cross at −1 and 1. ✓
Watch out. If two lines have the same y-intercept, use the gradient to tell them apart.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Match using the y-intercept first, then the sign of the gradient.
- Find the y-intercept of the equation. The number on its own, c, is where the line crosses the y-axis.y = 12x + 1: c = 1, so the line passes through (0, 1)
- Check the sign of the gradient. A positive m means the line goes up from left to right; a negative m means it goes down.m = 12 is positive: the line goes up
- Pick the line on the graph that matches both. Check the crossing of the y-axis and the direction.Line C crosses the y-axis at 1 and goes up.
Check. No other line crosses the y-axis at 1: the others cross at −1 and 3. ✓
Watch out. If two lines have the same y-intercept, use the gradient to tell them apart.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Match using the y-intercept first, then the sign of the gradient.
- Find the y-intercept of the equation. The number on its own, c, is where the line crosses the y-axis.y = 2x − 1: c = −1, so the line passes through (0, −1)
- Check the sign of the gradient. A positive m means the line goes up from left to right; a negative m means it goes down.m = 2 is positive: the line goes up
- Pick the line on the graph that matches both. Check the crossing of the y-axis and the direction.Line A crosses the y-axis at −1 and goes up.
Check. No other line crosses the y-axis at −1: the others cross at 3 and 1. ✓
Watch out. If two lines have the same y-intercept, use the gradient to tell them apart.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Start at the y-intercept, then use the gradient (rise ÷ run) to find more points. No table of values is needed.
- Find the y-intercept and mark it. The number on its own is c. The line crosses the y-axis at (0, c), so this is the first point.c = 1: mark (0, 1)
- Use the gradient to find a second point. The gradient is 2, which is 2/1: go 1 across and 2 up.from (0, 1) go 1 right and 2 up to (1, 3)
- Go the other way to get a third point as a check. Going left by the same amount means doing the opposite of the rise too.from (0, 1) go 1 left and 2 down to (−1, −1)
- Draw a straight line through the points with a ruler. Extend the line across the grid. Label it with its equation.
Check. Pick the point with x = 1: y = 2 × 1 + 1 = 3, which is on the sketch. ✓
Watch out. For a negative gradient go down as you go across (or up if you go left).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Start at the y-intercept, then use the gradient (rise ÷ run) to find more points. No table of values is needed.
- Find the y-intercept and mark it. The number on its own is c. The line crosses the y-axis at (0, c), so this is the first point.c = 3: mark (0, 3)
- Use the gradient to find a second point. The gradient is −1, which is −1/1: go 1 across and 1 down.from (0, 3) go 1 right and 1 down to (1, 2)
- Go the other way to get a third point as a check. Going left by the same amount means doing the opposite of the rise too.from (0, 3) go 1 left and 1 up to (−1, 4)
- Draw a straight line through the points with a ruler. Extend the line across the grid. Label it with its equation.
Check. Pick the point with x = 1: y = −1 + 3 = 2, which is on the sketch. ✓
Watch out. For a negative gradient go down as you go across (or up if you go left).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Start at the y-intercept, then use the gradient (rise ÷ run) to find more points. No table of values is needed.
- Find the y-intercept and mark it. The number on its own is c. The line crosses the y-axis at (0, c), so this is the first point.c = −2: mark (0, −2)
- Use the gradient to find a second point. The gradient is 12, which means: go 2 across and 1 up (rise ÷ run = 1/2).from (0, −2) go 2 right and 1 up to (2, −1)
- Go the other way to get a third point as a check. Going left by the same amount means doing the opposite of the rise too.from (0, −2) go 2 left and 1 down to (−2, −3)
- Draw a straight line through the points with a ruler. Extend the line across the grid. Label it with its equation.
Check. Pick the point with x = 2: y = 12 × 2 − 2 = −1, which is on the sketch. ✓
Watch out. For a negative gradient go down as you go across (or up if you go left).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Start at the y-intercept, then use the gradient (rise ÷ run) to find more points. No table of values is needed.
- Find the y-intercept and mark it. The number on its own is c. The line crosses the y-axis at (0, c), so this is the first point.c = 2: mark (0, 2)
- Use the gradient to find a second point. The gradient is −23, which means: go 3 across and 2 down (rise ÷ run = −2/3).from (0, 2) go 3 right and 2 down to (3, 0)
- Go the other way to get a third point as a check. Going left by the same amount means doing the opposite of the rise too.from (0, 2) go 3 left and 2 up to (−3, 4)
- Draw a straight line through the points with a ruler. Extend the line across the grid. Label it with its equation.
Check. Pick the point with x = 3: y = (−23) × 3 + 2 = 0, which is on the sketch. ✓
Watch out. For a negative gradient go down as you go across (or up if you go left).
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: −2x + y = 3
Add 2x to both sides: y = 2x + 3 - Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = 2, c = 3
Check. Test x = 3. Original: −2 × 3 + 1 × y = 3 gives y = 9. New form: y = 2 × 3 + 3 = 9. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: 2x + y = 7
Subtract 2x from both sides: y = −2x + 7 - Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = −2, c = 7
Check. Test x = 3. Original: 2 × 3 + 1 × y = 7 gives y = 1. New form: y = (−2) × 3 + 7 = 1. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Divide every term by the number in front of y to get the form y = mx + c.
- The equation has a number multiplying y. We want y on its own (coefficient 1), so divide both sides by that number.Start: 2y = 8x − 6
- Divide every term by 2. Every term on the right must be divided too, not just the first.y = 8x ÷ 2 − 6 ÷ 2
y = 4x − 3 - Read off the gradient and the y-intercept. Compare with y = mx + c.m = 4, c = −3
Check. Test x = 2: the original gives 2y = 10, so y = 5. The new equation gives y = 4 × 2 − 3 = 5. ✓
Watch out. Divide every term by the number in front of y, including the plain number.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Divide every term by the number in front of y to get the form y = mx + c.
- The equation has a number multiplying y. We want y on its own (coefficient 1), so divide both sides by that number.Start: 3y = 6x + 9
- Divide every term by 3. Every term on the right must be divided too, not just the first.y = 6x ÷ 3 + 9 ÷ 3
y = 2x + 3 - Read off the gradient and the y-intercept. Compare with y = mx + c.m = 2, c = 3
Check. Test x = 2: the original gives 3y = 21, so y = 7. The new equation gives y = 2 × 2 + 3 = 7. ✓
Watch out. Divide every term by the number in front of y, including the plain number.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: 3x + 4y = 12
Subtract 3x from both sides: 4y = −3x + 12 - Divide every term by the number in front of y. y must have a coefficient of 1. Divide every term (both of them) by 4.Divide every term on both sides by 4: y = −34x + 3
- Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = −34, c = 3
Check. Test x = 3. Original: 3 × 3 + 4 × y = 12 gives y = 0.75. New form: y = (−34) × 3 + 3 = 0.75. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: 5x − 2y = 10
Subtract 5x from both sides: −2y = −5x + 10 - Divide every term by the number in front of y. y must have a coefficient of 1. Divide every term (both of them) by −2.Divide every term on both sides by −2: y = 52x − 5
- Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = 52, c = −5
Check. Test x = 3. Original: 5 × 3 + (−2) × y = 10 gives y = 2.5. New form: y = 52 × 3 − 5 = 2.5. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: x + 3y = 9
Subtract 1x from both sides: 3y = −x + 9 - Divide every term by the number in front of y. y must have a coefficient of 1. Divide every term (both of them) by 3.Divide every term on both sides by 3: y = −13x + 3
- Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = −13, c = 3
Check. Test x = 3. Original: 1 × 3 + 3 × y = 9 gives y = 2. New form: y = (−13) × 3 + 3 = 2. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange to make y the subject. Whatever you do to one side you must do to the other side, and divide every term.
- Get the y term on its own on one side. We want the equation to start y = ..., so move everything except the y term to the other side using the opposite operation, on both sides.Start: 4x + 2y = −5
Subtract 4x from both sides: 2y = −4x − 5 - Divide every term by the number in front of y. y must have a coefficient of 1. Divide every term (both of them) by 2.Divide every term on both sides by 2: y = −2x − 2.5
- Read off the gradient and the intercept. Compare with y = mx + c: the number in front of x is m and the number on its own is c.m = −2, c = −52
Check. Test x = 3. Original: 4 × 3 + 2 × y = −5 gives y = −8.5. New form: y = (−2) × 3 − 2.5 = −8.5. They match. ✓
Watch out. When you divide by a number, divide every term, including the x term. Moving a term across changes its sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Substitute the coordinates of the point into the equation and solve for c.
- Write the equation with the known gradient. The line is y = 3x + c. Only c is unknown.y = 3x + c
- Substitute the point (2, 5). The point is on the line, so its coordinates make the equation true. Replace x and y by the numbers.5 = 3 × 2 + c
5 = 6 + c - Solve for c. Take 6 from both sides.c = 5 − 6 = −1
Check. Put c back: y = 3 × 2 − 1 = 5, which is 5. ✓
Watch out. Replace x and y with the point’s numbers: x first, y second.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Substitute the coordinates of the point into the equation and solve for c.
- Write the equation with the known gradient. The line is y = −2x + c. Only c is unknown.y = −2x + c
- Substitute the point (1, 4). The point is on the line, so its coordinates make the equation true. Replace x and y by the numbers.4 = (−2) × 1 + c
4 = −2 + c - Solve for c. Add 2 to both sides.c = 4 − (−2) = 6
Check. Put c back: y = (−2) × 1 + 6 = 4, which is 4. ✓
Watch out. Replace x and y with the point’s numbers: x first, y second.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Substitute the coordinates of the point into the equation and solve for c.
- Write the equation with the known gradient. The line is y = 12x + c. Only c is unknown.y = 12x + c
- Substitute the point (4, 3). The point is on the line, so its coordinates make the equation true. Replace x and y by the numbers.3 = 12 × 4 + c
3 = 2 + c - Solve for c. Take 2 from both sides.c = 3 − 2 = 1
Check. Put c back: y = 12 × 4 + 1 = 3, which is 3. ✓
Watch out. Replace x and y with the point’s numbers: x first, y second.
Key words. An intercept is a place where a line crosses an axis.
Method. Cross the y-axis: put x = 0. Cross the x-axis: put y = 0.
- The line crosses the y-axis where x = 0. On the y-axis every point has an x-coordinate of 0.y = 2 × 0 − 6 = −6
The y-intercept is (0, −6). - The line crosses the x-axis where y = 0. On the x-axis every point has a y-coordinate of 0. Put y = 0 into the equation.0 = 2x − 6
- Solve for x. Use inverse operations on both sides.Add 6 to both sides: 6 = 2x
Divide both sides by 2: x = 6 ÷ 2 = 3 - Write the intercepts as coordinates. x-intercept (3, 0) and y-intercept (0, −6)
Check. Substitute (3, 0): y = 2 × 3 − 6 = 0 = 0. ✓
Watch out. The x-intercept is where y = 0, not where x = 0. Do not mix them up.
Key words. An intercept is a place where a line crosses an axis.
Method. Cross the y-axis: put x = 0. Cross the x-axis: put y = 0.
- The line crosses the y-axis where x = 0. On the y-axis every point has an x-coordinate of 0.y = −0 + 4 = 4
The y-intercept is (0, 4). - The line crosses the x-axis where y = 0. On the x-axis every point has a y-coordinate of 0. Put y = 0 into the equation.0 = −x + 4
- Solve for x. Use inverse operations on both sides.Subtract 4 from both sides: −4 = −x
Divide both sides by −1: x = −4 ÷ (−1) = 4 - Write the intercepts as coordinates. x-intercept (4, 0) and y-intercept (0, 4)
Check. Substitute (4, 0): y = −4 + 4 = 0 = 0. ✓
Watch out. The x-intercept is where y = 0, not where x = 0. Do not mix them up.
Key words. An intercept is a place where a line crosses an axis.
Method. Cross the y-axis: put x = 0. Cross the x-axis: put y = 0.
- The line crosses the y-axis where x = 0. On the y-axis every point has an x-coordinate of 0.y = 12 × 0 + 3 = 3
The y-intercept is (0, 3). - The line crosses the x-axis where y = 0. On the x-axis every point has a y-coordinate of 0. Put y = 0 into the equation.0 = 12x + 3
- Solve for x. Use inverse operations on both sides.Subtract 3 from both sides: −3 = 12x
Divide both sides by 0.5: x = −3 ÷ 0.5 = −6 - Write the intercepts as coordinates. x-intercept (−6, 0) and y-intercept (0, 3)
Check. Substitute (−6, 0): y = 12 × (−6) + 3 = 0 = 0. ✓
Watch out. The x-intercept is where y = 0, not where x = 0. Do not mix them up.
Key words. An intercept is a place where a line crosses an axis.
Method. Cross the y-axis: put x = 0. Cross the x-axis: put y = 0.
- The line crosses the y-axis where x = 0. On the y-axis every point has an x-coordinate of 0.y = 3 × 0 + 12 = 12
The y-intercept is (0, 12). - The line crosses the x-axis where y = 0. On the x-axis every point has a y-coordinate of 0. Put y = 0 into the equation.0 = 3x + 12
- Solve for x. Use inverse operations on both sides.Subtract 12 from both sides: −12 = 3x
Divide both sides by 3: x = −12 ÷ 3 = −4 - Write the intercepts as coordinates. x-intercept (−4, 0) and y-intercept (0, 12)
Check. Substitute (−4, 0): y = 3 × (−4) + 12 = 0 = 0. ✓
Watch out. The x-intercept is where y = 0, not where x = 0. Do not mix them up.
Method. A line is y = mx + c. Write the equation using the gradient and the intercept, then test the point.
- Write the equation. Gradient 3 and y-intercept −4 (it passes through (0, −4)).y = 3x − 4
- Substitute x = 2. If the line passes through (2, 2), then y must be 2 when x = 2.y = 3 × 2 − 4 = 6 − 4 = 2
- Compare. The calculated y equals the y-coordinate of the point.2 = 2, so the line does pass through (2, 2).
Check. Going 2 across from (0, −4) at gradient 3 means going up 6, to (2, 2). ✓
Watch out. Substitute x and compare the y value: do not just look at the y-intercept.
Method. Parallel lines have the same gradient. Compare the gradients and the intercepts.
- Write down both gradients. The number multiplying x is the gradient.Both lines have gradient 2.
- Write down both y-intercepts. The number on its own.First line: 3. Second line: −1. They are different.
- Explain. Equal gradients mean the lines have the same steepness so they never get closer together or further apart. Different intercepts mean they are not the same line.The lines have the same gradient, so they are parallel. They have different y-intercepts, so they are different lines. Parallel lines never meet.
Check. Plot (0, 3), (1, 5) and (0, −1), (1, 1): both go up 2 for 1 across and stay 4 apart vertically. ✓
Watch out. “They never cross” needs a reason: say the gradients are equal.
Method. The sign belongs to the term after it. Rewrite the equation with the x term first.
- Rewrite in the form y = mx + c. Put the x term first and keep its sign.y = 3 − 2x = −2x + 3
- Read the gradient. The number multiplying x, with its sign.m = −2
- Read the y-intercept. The number on its own.c = 3
- Say what Jo got wrong. Jo used the first number as m and the second as c.Jo is wrong: the gradient is −2 and the y-intercept is 3.
Check. When x = 0, y = 3 − 0 = 3, which is the intercept. When x = 1, y = 1: it has gone down 2. ✓
Watch out. Do not read the numbers in the order they are written.
Key words. The x-intercept is where the line crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
Method. Find where the line crosses each axis by putting x = 0 and then y = 0. Join the two points.
- Find where the line crosses the y-axis: put x = 0. Every point on the y-axis has x = 0. The x term disappears.3y = 12 → y = 12 ÷ 3 = 4
The line crosses the y-axis at (0, 4). - Find where the line crosses the x-axis: put y = 0. Every point on the x-axis has y = 0. The y term disappears.2x = 12 → x = 12 ÷ 2 = 6
The line crosses the x-axis at (6, 0). - Plot the two points and join them with a ruler. Two points are enough for a straight line. Extend the line beyond them.
Check. Test a third point, such as x = 8: y = (12 − 2 × 8) ÷ 3 = −43, and the point is on the sketch. ✓
Watch out. For the y-intercept the x value is 0; for the x-intercept the y value is 0. Do not mix them up.
Key words. The x-intercept is where the line crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
Method. Find where the line crosses each axis by putting x = 0 and then y = 0. Join the two points.
- Find where the line crosses the y-axis: put x = 0. Every point on the y-axis has x = 0. The x term disappears.−2y = 10 → y = 10 ÷ (−2) = −5
The line crosses the y-axis at (0, −5). - Find where the line crosses the x-axis: put y = 0. Every point on the x-axis has y = 0. The y term disappears.5x = 10 → x = 10 ÷ 5 = 2
The line crosses the x-axis at (2, 0). - Plot the two points and join them with a ruler. Two points are enough for a straight line. Extend the line beyond them.
Check. Test a third point, such as x = 4: y = (10 − 5 × 4) ÷ (−2) = 5, and the point is on the sketch. ✓
Watch out. For the y-intercept the x value is 0; for the x-intercept the y value is 0. Do not mix them up.
Key words. The x-intercept is where the line crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
Method. Find where the line crosses each axis by putting x = 0 and then y = 0. Join the two points.
- Find where the line crosses the y-axis: put x = 0. Every point on the y-axis has x = 0. The x term disappears.4y = −12 → y = −12 ÷ 4 = −3
The line crosses the y-axis at (0, −3). - Find where the line crosses the x-axis: put y = 0. Every point on the x-axis has y = 0. The y term disappears.3x = −12 → x = −12 ÷ 3 = −4
The line crosses the x-axis at (−4, 0). - Plot the two points and join them with a ruler. Two points are enough for a straight line. Extend the line beyond them.
Check. Test a third point, such as x = −2: y = (−12 − 3 × (−2)) ÷ 4 = −1.5, and the point is on the sketch. ✓
Watch out. For the y-intercept the x value is 0; for the x-intercept the y value is 0. Do not mix them up.
Key words. The x-intercept is where the line crosses the x-axis (y = 0). The y-intercept is where it crosses the y-axis (x = 0).
Method. Find where the line crosses each axis by putting x = 0 and then y = 0. Join the two points.
- Find where the line crosses the y-axis: put x = 0. Every point on the y-axis has x = 0. The x term disappears.1y = 6 → y = 6 ÷ 1 = 6
The line crosses the y-axis at (0, 6). - Find where the line crosses the x-axis: put y = 0. Every point on the x-axis has y = 0. The y term disappears.2x = 6 → x = 6 ÷ 2 = 3
The line crosses the x-axis at (3, 0). - Plot the two points and join them with a ruler. Two points are enough for a straight line. Extend the line beyond them.
Check. Test a third point, such as x = 5: y = (6 − 2 × 5) ÷ 1 = −4, and the point is on the sketch. ✓
Watch out. For the y-intercept the x value is 0; for the x-intercept the y value is 0. Do not mix them up.
Method. The line crosses the axes at (3, 0) and (0, 6), so the triangle has a right angle at the origin with base 3 and height 6.
- Find the y-intercept. Put x = 0.y = −2 × 0 + 6 = 6, so (0, 6)
- Find the x-intercept. Put y = 0 and solve.0 = −2x + 6, so 2x = 6 and x = 3, giving (3, 0)
- Use the axes as base and height. The two axes meet at a right angle at the origin.base = 3, height = 6
- Work out the area. Area of a triangle = ½ × base × height.½ × 3 × 6 = 9
Check. The triangle is half of a 3 by 6 rectangle: 18 ÷ 2 = 9. ✓
Watch out. Use the intercepts as lengths: lengths must be positive.