GCSE Maths: Where lines cross (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)
Where lines cross
When two lines are drawn on the same grid, the point where they cross is called the point of intersection. It is the only point that lies on both lines, so its coordinates make both equations true. You can find it in three ways.
- Draw and read. Draw both lines accurately (from tables of values), then read the coordinates of the crossing point. Check your answer by substituting into both equations.
- Lines parallel to the axes. The line x = 3 meets y = 2x − 1 where x = 3; substitute to get y = 5, so the point is (3, 5). The line y = 7 meets it where 7 = 2x − 1, so x = 4. A line crosses the y-axis where x = 0 and the x-axis where y = 0.
- Algebra (Higher). At the crossing point the two y values are equal, so set the right-hand sides equal and solve for x, then find y. This leads to simultaneous equations in Chapter 14.
Two lines with the same gradient are parallel: they never cross, so there is no point of intersection. Two different straight lines can cross at most once.
Solving equations with a graph
A graph can also solve an equation. To solve 2x − 1 = 5, draw the line y = 2x − 1 and the horizontal line y = 5. They cross where 2x − 1 equals 5, and the x-coordinate of that point is the solution, x = 3. You can use this idea whenever you have an equation with the same expression on both sides and a graph of that expression.
Key ideas
- The point of intersection lies on both lines, so its coordinates satisfy both equations. Check by substituting into both.
- To find where a line meets x = a, put x = a; to find where it meets y = b, put y = b.
- To solve an equation graphically, draw y = (left side) and y = (right side); the x of the crossing point is the solution.
- Parallel lines (equal gradients) do not intersect.
- Higher: set the two expressions for y equal, solve for x, then substitute to find y.
Key words. The point of intersection of two lines is the point where they cross.
Method. Where two lines cross, the point is on both lines. Read its coordinates from the graph and check them in both equations.
- Find the point where the two lines cross. The point of intersection is the only point that is on both lines.
- Read its coordinates. Go across from the origin to find x, then up or down to find y.The lines cross at (3, 3).
- Check in both equations. The point must satisfy both equations if it is on both lines.y = 2x − 3: 2 × 3 − 3 = 3 ✓
y = −x + 6: −3 + 6 = 3 ✓
Check. Both equations give y = 3 when x = 3. ✓
Watch out. Give the answer as a coordinate pair (x, y), not just one 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. Draw both lines on one grid, then read the coordinates of the point where they cross.
- Make a table of values for y = x + 1. Substitute each x value. Put negatives in brackets.x = 0: y = 0 + 1 = 1
x = 2: y = 2 + 1 = 3
x = 4: y = 4 + 1 = 5
x = 6: y = 6 + 1 = 7x 0 2 4 6 y 1 3 5 7 - Make a table of values for y = −x + 7. Do the same for the second equation.x = 0: y = −0 + 7 = 0 + 7 = 7
x = 2: y = −2 + 7 = −2 + 7 = 5
x = 4: y = −4 + 7 = −4 + 7 = 3
x = 6: y = −6 + 7 = −6 + 7 = 1x 0 2 4 6 y 7 5 3 1 - Plot both lines on the same grid. Plot each set of points and join them with a ruler. Label each line.
- Read the point where the lines cross. This is the point that is on both lines.The lines cross at (3, 4).
Check. Check in both equations: y = x + 1 gives 3 + 1 = 4; y = −x + 7 gives −3 + 7 = 4. ✓
Watch out. Use a sharp pencil and a ruler. A thick or wobbly line makes the crossing point hard to read.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line y = 7 means y is always 7. At the crossing point both equations are true, so y = 7.y = 7
- Substitute into the other equation to find x. Replace y by 7 and solve.7 = 3x − 2
9 = 3x
x = 9 ÷ 3 = 3 - Write the coordinates. (3, 7)
Check. Put x = 3 in: y = 3 × 3 − 2 = 7. ✓
Watch out. Finish by writing the point as (x, y) with x first.
Key words. To solve an equation graphically means to use a graph to find the value of x that makes it true.
Method. Draw the horizontal line y = (the right-hand number). Where it meets the graph, read off x.
- Rewrite the equation as a question about the graph. Solving 2x − 1 = 5 means: for what x does the line y = 2x − 1 have y = 5?Draw the horizontal line y = 5.
- Find where the two lines cross. At the crossing point the two y values are equal, so the x value solves the equation.
- Read the x-coordinate of the crossing point. Go straight down from the crossing point to the x-axis.x = 3
Check. Substitute: 2 × 3 − 1 = 5 = 5. ✓
Watch out. The answer is the x-coordinate, not the y-coordinate of the crossing point.
Key words. The point of intersection is where two lines cross. Its coordinates fit both equations.
Method. At the point of intersection both equations are true, so the two expressions for y are equal. Solve for x, then find y.
- Both equations give y, so set the right-hand sides equal. At the point where the lines cross, x and y are the same for both lines.2x + 1 = −x + 7
- Get the x terms on one side. Collect the x terms on one side and the numbers on the other, doing the same to both sides.3x = 6 (the x terms: 2x − (−1)x = 3x; the numbers: 7 − 1 = 6)
- Solve for x. Divide both sides by 3.x = 6 ÷ 3 = 2
- Find y by substituting x into either equation. Use the easier one.y = 2 × 2 + 1 = 5
- Write the point of intersection. (2, 5)
Check. Put x = 2 into the other equation: y = −2 + 7 = 5. Both give 5. ✓
Watch out. Remember to find y as well: the answer is a coordinate pair, not just a value of x.
Watch out
Give a coordinate pair. The intersection is a point, so write (x, y), not just x. And always check in both equations: if only one works, the crossing point has been misread. When solving an equation graphically, the answer is the x-coordinate of the crossing point, not the y-coordinate.Exercise 12M Intersections of lines
Non-calculatorKey words. The point of intersection of two lines is the point where they cross.
Method. Where two lines cross, the point is on both lines. Read its coordinates from the graph and check them in both equations.
- Find the point where the two lines cross. The point of intersection is the only point that is on both lines.
- Read its coordinates. Go across from the origin to find x, then up or down to find y.The lines cross at (2, 3).
- Check in both equations. The point must satisfy both equations if it is on both lines.y = x + 1: 2 + 1 = 3 ✓
y = −x + 5: −2 + 5 = 3 ✓
Check. Both equations give y = 3 when x = 2. ✓
Watch out. Give the answer as a coordinate pair (x, y), not just one number.
Key words. The point of intersection of two lines is the point where they cross.
Method. Where two lines cross, the point is on both lines. Read its coordinates from the graph and check them in both equations.
- Find the point where the two lines cross. The point of intersection is the only point that is on both lines.
- Read its coordinates. Go across from the origin to find x, then up or down to find y.The lines cross at (3, 3).
- Check in both equations. The point must satisfy both equations if it is on both lines.y = 2x − 3: 2 × 3 − 3 = 3 ✓
y = −x + 6: −3 + 6 = 3 ✓
Check. Both equations give y = 3 when x = 3. ✓
Watch out. Give the answer as a coordinate pair (x, y), not just one number.
Key words. The point of intersection of two lines is the point where they cross.
Method. Where two lines cross, the point is on both lines. Read its coordinates from the graph and check them in both equations.
- Find the point where the two lines cross. The point of intersection is the only point that is on both lines.
- Read its coordinates. Go across from the origin to find x, then up or down to find y.The lines cross at (2, 2).
- Check in both equations. The point must satisfy both equations if it is on both lines.y = x/2 + 1: 12 × 2 + 1 = 2 ✓
y = −x + 4: −2 + 4 = 2 ✓
Check. Both equations give y = 2 when x = 2. ✓
Watch out. Give the answer as a coordinate pair (x, y), not just one 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. Draw both lines on one grid, then read the coordinates of the point where they cross.
- Make a table of values for y = x + 2. Substitute each x value. Put negatives in brackets.x = −1: y = (−1) + 2 = 1
x = 0: y = 0 + 2 = 2
x = 1: y = 1 + 2 = 3
x = 2: y = 2 + 2 = 4
x = 3: y = 3 + 2 = 5
x = 4: y = 4 + 2 = 6x −1 0 1 2 3 4 y 1 2 3 4 5 6 - Make a table of values for y = −x + 4. Do the same for the second equation.x = −1: y = −(−1) + 4 = 1 + 4 = 5
x = 0: y = −0 + 4 = 0 + 4 = 4
x = 1: y = −1 + 4 = −1 + 4 = 3
x = 2: y = −2 + 4 = −2 + 4 = 2
x = 3: y = −3 + 4 = −3 + 4 = 1
x = 4: y = −4 + 4 = −4 + 4 = 0x −1 0 1 2 3 4 y 5 4 3 2 1 0 - Plot both lines on the same grid. Plot each set of points and join them with a ruler. Label each line.
- Read the point where the lines cross. This is the point that is on both lines.The lines cross at (1, 3).
Check. Check in both equations: y = x + 2 gives 1 + 2 = 3; y = −x + 4 gives −1 + 4 = 3. ✓
Watch out. Use a sharp pencil and a ruler. A thick or wobbly line makes the crossing point hard to read.
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. Draw both lines on one grid, then read the coordinates of the point where they cross.
- Make a table of values for y = 2x − 1. Substitute each x value. Put negatives in brackets.x = 0: y = 2 × 0 − 1 = 0 − 1 = −1
x = 1: y = 2 × 1 − 1 = 2 − 1 = 1
x = 2: y = 2 × 2 − 1 = 4 − 1 = 3
x = 3: y = 2 × 3 − 1 = 6 − 1 = 5
x = 4: y = 2 × 4 − 1 = 8 − 1 = 7x 0 1 2 3 4 y −1 1 3 5 7 - Make a table of values for y = −x + 5. Do the same for the second equation.x = 0: y = −0 + 5 = 0 + 5 = 5
x = 1: y = −1 + 5 = −1 + 5 = 4
x = 2: y = −2 + 5 = −2 + 5 = 3
x = 3: y = −3 + 5 = −3 + 5 = 2
x = 4: y = −4 + 5 = −4 + 5 = 1x 0 1 2 3 4 y 5 4 3 2 1 - Plot both lines on the same grid. Plot each set of points and join them with a ruler. Label each line.
- Read the point where the lines cross. This is the point that is on both lines.The lines cross at (2, 3).
Check. Check in both equations: y = 2x − 1 gives 2 × 2 − 1 = 3; y = −x + 5 gives −2 + 5 = 3. ✓
Watch out. Use a sharp pencil and a ruler. A thick or wobbly line makes the crossing point hard to read.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line x = 0 means x is always 0. At the crossing point both equations are true, so x = 0.x = 0
- Substitute into the other equation to find y. Replace x by 0.y = 2 × 0 + 1 = 1
- Write the coordinates. (0, 1)
Check. The point (0, 1) has x = 0, so it is on the vertical line. ✓
Watch out. Finish by writing the point as (x, y) with x first.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line y = 0 means y is always 0. At the crossing point both equations are true, so y = 0.y = 0
- Substitute into the other equation to find x. Replace y by 0 and solve.0 = 2x + 1
−1 = 2x
x = −1 ÷ 2 = −0.5 - Write the coordinates. (−0.5, 0)
Check. Put x = −0.5 in: y = 2 × (−0.5) + 1 = 0. ✓
Watch out. Finish by writing the point as (x, y) with x first.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line x = 3 means x is always 3. At the crossing point both equations are true, so x = 3.x = 3
- Substitute into the other equation to find y. Replace x by 3.y = 2 × 3 + 1 = 7
- Write the coordinates. (3, 7)
Check. The point (3, 7) has x = 3, so it is on the vertical line. ✓
Watch out. Finish by writing the point as (x, y) with x first.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line y = −3 means y is always −3. At the crossing point both equations are true, so y = −3.y = −3
- Substitute into the other equation to find x. Replace y by −3 and solve.−3 = 2x + 1
−4 = 2x
x = −4 ÷ 2 = −2 - Write the coordinates. (−2, −3)
Check. Put x = −2 in: y = 2 × (−2) + 1 = −3. ✓
Watch out. Finish by writing the point as (x, y) with x first.
Key words. To solve an equation graphically means to use a graph to find the value of x that makes it true.
Method. Draw the horizontal line y = (the right-hand number). Where it meets the graph, read off x.
- Rewrite the equation as a question about the graph. Solving 2x − 1 = 5 means: for what x does the line y = 2x − 1 have y = 5?Draw the horizontal line y = 5.
- Find where the two lines cross. At the crossing point the two y values are equal, so the x value solves the equation.
- Read the x-coordinate of the crossing point. Go straight down from the crossing point to the x-axis.x = 3
Check. Substitute: 2 × 3 − 1 = 5 = 5. ✓
Watch out. The answer is the x-coordinate, not the y-coordinate of the crossing point.
Key words. To solve an equation graphically means to use a graph to find the value of x that makes it true.
Method. Draw the horizontal line y = (the right-hand number). Where it meets the graph, read off x.
- Rewrite the equation as a question about the graph. Solving 2x − 1 = −3 means: for what x does the line y = 2x − 1 have y = −3?Draw the horizontal line y = −3.
- Find where the two lines cross. At the crossing point the two y values are equal, so the x value solves the equation.
- Read the x-coordinate of the crossing point. Go straight down from the crossing point to the x-axis.x = −1
Check. Substitute: 2 × (−1) − 1 = −3 = −3. ✓
Watch out. The answer is the x-coordinate, not the y-coordinate of the crossing point.
Key words. To solve an equation graphically means to use a graph to find the value of x that makes it true.
Method. Draw the horizontal line y = (the right-hand number). Where it meets the graph, read off x.
- Rewrite the equation as a question about the graph. Solving 2x − 1 = 0 means: for what x does the line y = 2x − 1 have y = 0?Draw the horizontal line y = 0.
- Find where the two lines cross. At the crossing point the two y values are equal, so the x value solves the equation.
- Read the x-coordinate of the crossing point. Go straight down from the crossing point to the x-axis.x = 0.5
Check. Substitute: 2 × 0.5 − 1 = 0 = 0. ✓
Watch out. The answer is the x-coordinate, not the y-coordinate of the crossing point.
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. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 3 × 2 − 1 = 5
- Compare with the y-coordinate of the point. The point is (2, 5), so its y-coordinate is 5.5 = 5
- Write the conclusion. The point is on the line only if the two values are equal.(2, 5) is on the line y = 3x − 1.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
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. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 2 + 3 = 5
- Compare with the y-coordinate of the point. The point is (2, 5), so its y-coordinate is 5.5 = 5
- Write the conclusion. The point is on the line only if the two values are equal.(2, 5) is on the line y = x + 3.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Method. A point of intersection is on both lines.
- Use parts (a) and (b). P is on the first line and on the second line.Yes and Yes
- Conclude. A point that lies on both lines is where they cross.P(2, 5) is the point of intersection.
Check. Two different lines cross at only one point, so P is the only crossing point. ✓
Watch out. You must check both equations.
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. At the point where two lines cross, both equations are true. Use the simple equation to give one coordinate, then substitute to find the other.
- The line x = −2 means x is always −2. At the crossing point both equations are true, so x = −2.x = −2
- Substitute into the other equation to find y. Replace x by −2.y = 12 × (−2) + 3 = 2
- Write the coordinates. (−2, 2)
Check. The point (−2, 2) has x = −2, so it is on the vertical line. ✓
Watch out. Finish by writing the point as (x, y) with x first.
Method. Parallel lines have the same gradient. Look at the gradients.
- Write the gradients. The number in front of x.y = 3x + 1 has m = 3; y = 3x − 4 has m = 3
- Compare. Equal gradients mean the lines have the same slope.They are parallel.
- Explain why they do not cross. Parallel lines stay the same distance apart.They never meet, so there is no point of intersection.
Check. Setting 3x + 1 = 3x − 4 would give 1 = −4, which is impossible. ✓
Watch out. Different intercepts mean they are not the same line.
Method. Use the fact that exactly one straight line goes through any two different points.
- Suppose two different lines met at two points. Call the points P and Q.Both lines would pass through P and Q.
- Use what we know about lines. Only one straight line can be drawn through two given points.So the two lines would be the same line.
- Conclude. This contradicts “different lines”.Two different straight lines can meet at most once.
Check. Try drawing two straight lines that cross twice: it cannot be done. ✓
Watch out. Give a reason, not just “it is obvious”.
Key words. The point of intersection is where two lines cross. Its coordinates fit both equations.
Method. At the point of intersection both equations are true, so the two expressions for y are equal. Solve for x, then find y.
- Both equations give y, so set the right-hand sides equal. At the point where the lines cross, x and y are the same for both lines.2x + 1 = −x + 7
- Get the x terms on one side. Collect the x terms on one side and the numbers on the other, doing the same to both sides.3x = 6 (the x terms: 2x − (−1)x = 3x; the numbers: 7 − 1 = 6)
- Solve for x. Divide both sides by 3.x = 6 ÷ 3 = 2
- Find y by substituting x into either equation. Use the easier one.y = 2 × 2 + 1 = 5
- Write the point of intersection. (2, 5)
Check. Put x = 2 into the other equation: y = −2 + 7 = 5. Both give 5. ✓
Watch out. Remember to find y as well: the answer is a coordinate pair, not just a value of x.
Key words. The point of intersection is where two lines cross. Its coordinates fit both equations.
Method. At the point of intersection both equations are true, so the two expressions for y are equal. Solve for x, then find y.
- Both equations give y, so set the right-hand sides equal. At the point where the lines cross, x and y are the same for both lines.3x − 2 = x + 4
- Get the x terms on one side. Collect the x terms on one side and the numbers on the other, doing the same to both sides.2x = 6 (the x terms: 3x − 1x = 2x; the numbers: 4 − (−2) = 6)
- Solve for x. Divide both sides by 2.x = 6 ÷ 2 = 3
- Find y by substituting x into either equation. Use the easier one.y = 3 × 3 − 2 = 7
- Write the point of intersection. (3, 7)
Check. Put x = 3 into the other equation: y = 3 + 4 = 7. Both give 7. ✓
Watch out. Remember to find y as well: the answer is a coordinate pair, not just a value of x.
Key words. The point of intersection is where two lines cross. Its coordinates fit both equations.
Method. At the point of intersection both equations are true, so the two expressions for y are equal. Solve for x, then find y.
- Both equations give y, so set the right-hand sides equal. At the point where the lines cross, x and y are the same for both lines.4x − 5 = −2x + 7
- Get the x terms on one side. Collect the x terms on one side and the numbers on the other, doing the same to both sides.6x = 12 (the x terms: 4x − (−2)x = 6x; the numbers: 7 − (−5) = 12)
- Solve for x. Divide both sides by 6.x = 12 ÷ 6 = 2
- Find y by substituting x into either equation. Use the easier one.y = 4 × 2 − 5 = 3
- Write the point of intersection. (2, 3)
Check. Put x = 2 into the other equation: y = (−2) × 2 + 7 = 3. Both give 3. ✓
Watch out. Remember to find y as well: the answer is a coordinate pair, not just a value of x.
Key words. The point of intersection is where two lines cross. Its coordinates fit both equations.
Method. At the point of intersection both equations are true, so the two expressions for y are equal. Solve for x, then find y.
- Both equations give y, so set the right-hand sides equal. At the point where the lines cross, x and y are the same for both lines.12x + 1 = −x + 4
- Get the x terms on one side. Collect the x terms on one side and the numbers on the other, doing the same to both sides.32x = 3 (the x terms: 0.5x − (−1)x = 1.5x; the numbers: 4 − 1 = 3)
- Solve for x. Divide both sides by 1.5.x = 3 ÷ 1.5 = 2
- Find y by substituting x into either equation. Use the easier one.y = 12 × 2 + 1 = 2
- Write the point of intersection. (2, 2)
Check. Put x = 2 into the other equation: y = −2 + 4 = 2. Both give 2. ✓
Watch out. Remember to find y as well: the answer is a coordinate pair, not just a value of x.
Method. Find the three corners of the triangle, then use area = ½ × base × height.
- Draw the lines. The triangle is bounded by the two lines and the x-axis (y = 0).
- Corner where the first line meets the x-axis. Put y = 0 in y = x.(0, 0)
- Corner where the second line meets the x-axis. Put y = 0 in y = −x + 6: 0 = −x + 6, so x = 6.(6, 0)
- Corner where the two lines cross. Set x = −x + 6, so 2x = 6 and x = 3, y = 3.(3, 3)
- Base and height. The base is along the x-axis from 0 to 6; the height is the y-coordinate of the top corner.base = 6, height = 3
- Area. ½ × 6 × 3 = 9
Check. The triangle is half of a 6 by 3 rectangle: 18 ÷ 2 = 9. ✓
Watch out. The height is perpendicular to the base: it is the y-coordinate of the top vertex.