GCSE Maths: Higher: the subject appears twice (Higher)

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the terms containing the subject on one side, factorise the subject out, then divide.

1. Write the formula. The subject x appears on both sides, so we cannot undo one operation at a time.
   px + q = rx + s
2. Collect every term containing x on the left and every other term on the right. Subtract rx from both sides and subtract q from both sides. Each time a term crosses the equals sign its sign changes.
   px − rx = s − q
3. Factorise the left side. Both terms contain x. Take x outside a bracket and put what is left inside.
   x(p − r) = s − q
4. Divide both sides by the bracket. x is multiplied by (p − r), so divide both sides by it.
   x = s − qp − r

Check. Pick numbers for the other letters: p = 6, q = 2, r = 5, s = 9. The new formula gives
x = 9 − 26 − 5 = 71 = 7
Now put x = 7 back into the original formula. Left side: 6 × 7 + 2 = 42 + 2 = 44. Right side: 5 × 7 + 9 = 35 + 9 = 44. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the terms containing the subject on one side, factorise the subject out, then divide.

1. Write the formula. The subject x appears on both sides.
   b + ax = d + cx
2. Collect the x terms on the left and the other terms on the right. Subtract cx and b from both sides.
   ax − cx = d − b
3. Factorise the left side. Take x outside a bracket.
   x(a − c) = d − b
4. Divide both sides by the bracket. Divide by (a − c).
   x = d − ba − c

Check. Pick numbers for the other letters: b = 3, a = 8, d = 9, c = 2. The new formula gives
x = 9 − 38 − 2 = 66 = 1
Now put x = 1 back into the original formula. Left side: 3 + 8 × 1 = 3 + 8 = 11. Right side: 9 + 2 × 1 = 9 + 2 = 11. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the terms containing the subject on one side, factorise the subject out, then divide.

1. Write the formula. The subject x appears on both sides, so we cannot undo one operation at a time.
   mx + k = nx + t
2. Collect every term containing x on the left and every other term on the right. Subtract nx from both sides and subtract k from both sides. Each time a term crosses the equals sign its sign changes.
   mx − nx = t − k
3. Factorise the left side. Both terms contain x. Take x outside a bracket and put what is left inside.
   x(m − n) = t − k
4. Divide both sides by the bracket. x is multiplied by (m − n), so divide both sides by it.
   x = t − km − n

Check. Pick numbers for the other letters: m = 6, k = 4, n = 5, t = 9. The new formula gives
x = 9 − 46 − 5 = 51 = 5
Now put x = 5 back into the original formula. Left side: 6 × 5 + 4 = 30 + 4 = 34. Right side: 5 × 5 + 9 = 25 + 9 = 34. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Expand the brackets, collect the terms containing the subject, factorise, then divide.

1. Write the formula. x is inside brackets on both sides.
   p(x + q) = r(x + s)
2. Expand both brackets. Multiply everything inside each bracket by the number outside.
   px + pq = rx + rs
3. Collect the x terms on the left and the others on the right. Subtract rx and pq from both sides.
   px − rx = rs − pq
4. Factorise the left side. Take x outside a bracket.
   x(p − r) = rs − pq
5. Divide both sides by the bracket. Divide by (p − r).
   x = rs − pqp − r

Check. Pick numbers for the other letters: p = 2, q = 10, r = 4, s = 3. The new formula gives
x = 4 × 3 − 2 × 102 − 4 = 12 − 20−2 = −8−2 = 4
Now put x = 4 back into the original formula. Left side: 2 × (4 + 10) = 2 × 14 = 28. Right side: 4 × (4 + 3) = 4 × 7 = 28. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Expand the brackets, collect the terms containing the subject, factorise, then divide.

1. Write the formula. x is inside brackets on both sides.
   a(x + b) = c(x + d)
2. Expand both brackets. Multiply everything inside each bracket by the number outside.
   ax + ab = cx + cd
3. Collect the x terms on the left and the others on the right. Subtract cx and ab from both sides.
   ax − cx = cd − ab
4. Factorise the left side. Take x outside a bracket.
   x(a − c) = cd − ab
5. Divide both sides by the bracket. Divide by (a − c).
   x = cd − aba − c

Check. Pick numbers for the other letters: a = 8, b = 3, c = 6, d = 5. The new formula gives
x = 6 × 5 − 8 × 38 − 6 = 30 − 242 = 62 = 3
Now put x = 3 back into the original formula. Left side: 8 × (3 + 3) = 8 × 6 = 48. Right side: 6 × (3 + 5) = 6 × 8 = 48. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and on the bottom of the fraction.
   y = ax + bx
2. Multiply both sides by x to remove the fraction. Multiplying by the bottom gets rid of the division. Do it to the whole of each side.
   yx = ax + b
3. Subtract ax from both sides. We want all the x terms on one side.
   yx − ax = b
4. Factorise the left side. Both terms contain x.
   x(y − a) = b
5. Divide both sides by the bracket. Divide by (y − a).
   x = by − a

Check. Pick numbers for the other letters: y = 4, a = 3, b = 2. The new formula gives
x = 24 − 3 = 21 = 2
Now put x = 2 back into the original formula. Left side: 4. Right side: 3 × 2 + 22 = 6 + 22 = 82 = 4. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and on the bottom of the fraction.
   y = px + qx
2. Multiply both sides by x to remove the fraction. Multiplying by the bottom gets rid of the division. Do it to the whole of each side.
   yx = px + q
3. Subtract px from both sides. We want all the x terms on one side.
   yx − px = q
4. Factorise the left side. Both terms contain x.
   x(y − p) = q
5. Divide both sides by the bracket. Divide by (y − p).
   x = qy − p

Check. Pick numbers for the other letters: y = 5, p = 2, q = 6. The new formula gives
x = 65 − 2 = 63 = 2
Now put x = 2 back into the original formula. Left side: 5. Right side: 2 × 2 + 62 = 4 + 62 = 102 = 5. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Factorise the subject out of the terms that contain it, then divide.

1. Write the formula. x appears twice: once by itself and once multiplied by k.
   T = x + kx
2. Factorise the right side. Both terms contain x. Take it outside a bracket: x is the same as 1 × x.
   T = x(1 + k)
3. Divide both sides by the bracket. Divide by (1 + k).
   T1 + k = x
4. Write with the subject on the left.
   x = T1 + k

Check. Pick numbers for the other letters: T = 3, k = 2. The new formula gives
x = 31 + 2 = 33 = 1
Now put x = 1 back into the original formula. Left side: 3. Right side: 1 + 2 × 1 = 1 + 2 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and the bottom of the fraction.
   y = x + 3x − 2
2. Multiply both sides by the bottom, (x − 2). This removes the fraction. The whole of the bottom is multiplied.
   y(x − 2) = x + 3
3. Expand the bracket. Multiply both terms inside by y.
   yx − 2y = x + 3
4. Collect the x terms on the left and the others on the right. Subtract x from both sides and add 2y to both sides.
   yx − x = 3 + 2y
5. Factorise the left side. Take x outside a bracket.
   x(y − 1) = 3 + 2y
6. Divide both sides by the bracket. Divide by (y − 1).
   x = 3 + 2yy − 1

Check. Pick numbers for the other letters: y = 6. The new formula gives
x = 3 + 2 × 66 − 1 = 3 + 125 = 155 = 3
Now put x = 3 back into the original formula. Left side: 6. Right side: 3 + 33 − 2 = 61 = 6. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and the bottom of the fraction.
   y = x + 5x − 1
2. Multiply both sides by the bottom, (x − 1). This removes the fraction. The whole of the bottom is multiplied.
   y(x − 1) = x + 5
3. Expand the bracket. Multiply both terms inside by y.
   yx − 1y = x + 5
4. Collect the x terms on the left and the others on the right. Subtract x from both sides and add 1y to both sides.
   yx − x = 5 + 1y
5. Factorise the left side. Take x outside a bracket.
   x(y − 1) = 5 + 1y
6. Divide both sides by the bracket. Divide by (y − 1).
   x = 5 + 1yy − 1

Check. Pick numbers for the other letters: y = 3. The new formula gives
x = 5 + 1 × 33 − 1 = 5 + 32 = 82 = 4
Now put x = 4 back into the original formula. Left side: 3. Right side: 4 + 54 − 1 = 93 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and the bottom of the fraction.
   y = x + px − q
2. Multiply both sides by the bottom, (x − q). This removes the fraction. The whole of the bottom is multiplied.
   y(x − q) = x + p
3. Expand the bracket. Multiply both terms inside by y.
   yx − yq = x + p
4. Collect the x terms on the left and the others on the right. Subtract x from both sides and add yq to both sides.
   yx − x = p + yq
5. Factorise the left side. Take x outside a bracket.
   x(y − 1) = p + yq
6. Divide both sides by the bracket. Divide by (y − 1).
   x = p + yqy − 1

Check. Pick numbers for the other letters: y = 5, p = 10, q = 6. The new formula gives
x = 10 + 5 × 65 − 1 = 10 + 304 = 404 = 10
Now put x = 10 back into the original formula. Left side: 5. Right side: 10 + 1010 − 6 = 204 = 5. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and the bottom of the fraction.
   y = ax + bcx + d
2. Multiply both sides by the bottom. This removes the fraction.
   y(cx + d) = ax + b
3. Expand the bracket. Multiply both terms inside by y.
   ycx + yd = ax + b
4. Collect the x terms on the left and the others on the right. Subtract ax and yd from both sides.
   ycx − ax = b − yd
5. Factorise the left side. Take x outside a bracket.
   x(yc − a) = b − yd
6. Divide both sides by the bracket. Divide by the whole bracket.
   x = b − ydyc − a

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect the subject terms, factorise, then divide.

1. Write the formula. x appears on the top and the bottom of the fraction.
   y = px + qrx + s
2. Multiply both sides by the bottom. This removes the fraction.
   y(rx + s) = px + q
3. Expand the bracket. Multiply both terms inside by y.
   yrx + ys = px + q
4. Collect the x terms on the left and the others on the right. Subtract px and ys from both sides.
   yrx − px = q − ys
5. Factorise the left side. Take x outside a bracket.
   x(yr − p) = q − ys
6. Divide both sides by the bracket. Divide by the whole bracket.
   x = q − ysyr − p

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the squared terms, factorise, divide, then take the square root of the whole side.

1. Write the formula. x² appears on both sides. Treat x² as the thing to collect.
   ax² + b = cx² + d
2. Collect the x² terms on the left and the others on the right. Subtract cx² and b from both sides.
   ax² − cx² = d − b
3. Factorise the left side. Take x² outside a bracket.
   x²(a − c) = d − b
4. Divide both sides by the bracket. Divide by (a − c).
   x² = d − ba − c
5. Take the square root of both sides. The inverse of squaring. Take the positive root (lengths are positive).
   x = √d − ba − c

Check. Pick numbers for the other letters: a = 6, b = 10, c = 8, d = 2. The new formula gives
x = √2 − 106 − 8 = √−8−2 = √4 = 2
Now put x = 2 back into the original formula. Left side: 6 × 2² + 10 = 6 × 4 + 10 = 24 + 10 = 34. Right side: 8 × 2² + 2 = 8 × 4 + 2 = 32 + 2 = 34. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the squared terms, factorise, divide, then take the square root of the whole side.

1. Write the formula. x² appears on both sides. Treat x² as the thing to collect.
   px² + q = rx² + s
2. Collect the x² terms on the left and the others on the right. Subtract rx² and q from both sides.
   px² − rx² = s − q
3. Factorise the left side. Take x² outside a bracket.
   x²(p − r) = s − q
4. Divide both sides by the bracket. Divide by (p − r).
   x² = s − qp − r
5. Take the square root of both sides. The inverse of squaring. Take the positive root (lengths are positive).
   x = √s − qp − r

Check. Pick numbers for the other letters: p = 9, q = 4, r = 10, s = 3. The new formula gives
x = √3 − 49 − 10 = √−1−1 = √1 = 1
Now put x = 1 back into the original formula. Left side: 9 × 1² + 4 = 9 × 1 + 4 = 9 + 4 = 13. Right side: 10 × 1² + 3 = 10 × 1 + 3 = 10 + 3 = 13. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the terms containing the subject on one side, factorise the subject out, then divide.

1. Write the formula. The subject n appears on both sides.
   a + bn = c + dn
2. Collect the n terms on the left and the other terms on the right. Subtract dn and a from both sides.
   bn − dn = c − a
3. Factorise the left side. Take n outside a bracket.
   n(b − d) = c − a
4. Divide both sides by the bracket. Divide by (b − d).
   n = c − ab − d

Check. Pick numbers for the other letters: a = 2, b = 5, c = 9, d = 4. The new formula gives
n = 9 − 25 − 4 = 71 = 7
Now put n = 7 back into the original formula. Left side: 2 + 5 × 7 = 2 + 35 = 37. Right side: 9 + 4 × 7 = 9 + 28 = 37. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Collect the terms containing the subject on one side, factorise the subject out, then divide.

1. Write the formula. The subject n appears on both sides.
   a + bn = c + dn
2. Collect the n terms on the left and the other terms on the right. Subtract dn and a from both sides.
   bn − dn = c − a
3. Factorise the left side. Take n outside a bracket.
   n(b − d) = c − a
4. Divide both sides by the bracket. Divide by (b − d).
   n = c − ab − d

Check. Pick numbers for the other letters: a = 2, b = 5, c = 9, d = 4. The new formula gives
n = 9 − 25 − 4 = 71 = 7
Now put n = 7 back into the original formula. Left side: 2 + 5 × 7 = 2 + 35 = 37. Right side: 9 + 4 × 7 = 9 + 28 = 37. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   n = c − ab − d
2. Substitute: c = 12, a = 30, b = 4, d = 7. Substitute means replace each letter with its number.
   n = 12 − 304 − 7
3. Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   12 − 30 = −18
   4 − 7 = −3
   n = −18−3
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   −18−3 means −18 ÷ −3 = 6 (a negative divided by a negative gives a positive)
   n = 6
5. Write the answer with its units. The answer should say what was being worked out.
   n = 6 miles

Check. Put the numbers straight into a calculator: (12 − 30)/(4 − 7) gives 6. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. When the subject (the letter we want) appears twice, we cannot simply undo operations one at a time. Instead we collect every term containing the subject on one side, factorise it out (write it as a product, with the subject outside a bracket), and then divide.

Method. Clear the fraction, expand, collect, factorise and divide.

1. Write the formula. Both sides contain x.
   y = x + 1x − 1
2. Multiply both sides by (x − 1). Clear the fraction.
   y(x − 1) = x + 1
3. Expand. Multiply out the bracket.
   yx − y = x + 1
4. Collect the x terms on the left. Subtract x and add y to both sides.
   yx − x = 1 + y
5. Factorise. Take x outside.
   x(y − 1) = 1 + y
6. Divide. By (y − 1).
   x = 1 + yy − 1

Check. Pick numbers for the other letters: y = 3. The new formula gives
x = 1 + 33 − 1 = 42 = 2
Now put x = 2 back into the original formula. Left side: 3. Right side: 2 + 12 − 1 = 31 = 3. Both sides are equal, so the rearranging is correct. ✓

Watch out. Terms with the subject go on one side and all other terms on the other. When you move a term across the equals sign, its sign changes. Remember to multiply every term when you clear a fraction.

Method. Compare the two formulae and test them with a number.

1. Compare. The original is y = (x + 1)/(x − 1). The new one is x = (y + 1)/(y − 1).
   They have exactly the same form with x and y swapped.
2. Test with x = 3. Work out y, then use the new formula.
   y = 3 + 13 − 1 = 42 = 2
   x = 2 + 12 − 1 = 31 = 3
3. Explain. Applying the formula twice takes you back to where you started.
   The formula is its own inverse.

Check. 3 goes to 2 and 2 goes back to 3. ✓

Watch out. Notice patterns, then test with numbers.