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This page sets out the method for changing the subject of a formula in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Changing the subject of a formula topic page.
When the new subject appears once, list what is being done to it and undo in reverse order.
When the subject is squared or under a root, sort out the rest first and leave the root/square to the end.
The last step done to x is the first one you undo.
To make x the subject of y = 2x + 6, subtract 6 from both sides, then divide by 2. This gives x = (y - 6)/2.
To make x the subject of y = 2x - 5, add 5 to both sides, then divide by 2. This gives x = (y + 5)/2. You add the 5 back.
To make x the subject of y = x/4, multiply both sides by 4. This gives x = 4y.
To make x the subject of y = 5 - x, add x to both sides and subtract y from both sides. This gives x = 5 - y.
Write the inverse of each operation in reverse order.
Read the formula as a machine that builds the old subject from the new one. In y=3x-7, x is multiplied by 3 and then 7 is subtracted. To get x alone, undo the last operation first: add 7, so y+7=3x. Then divide by 3: x=(y+7)/3. Dividing first also works, but you must divide every term: y/3=x-7/3, so x=y/3+7/3. Writing y/3-7 divides one term only, which is a different machine.
Write one operation per line, and show both sides. Each correct line is usually a method mark. Keep a fraction bar long enough that it is obvious what is being divided. x=(y+7)/3 is not the same as x=y/3+7. The bar is part of the algebra, not decoration. If a line gets crowded, start a new one rather than shrinking the bar until the 3 looks as if it only divides the 7.
Method: 1. Read the formula as the order of operations that built the current subject. 2. In y=3x-7, the new subject was multiplied by 3 and then 7 was subtracted. 3. Undo the last operation first: add 7 to both sides to get y+7=3x.
If the new subject is squared, isolate that square and then square-root both sides. From A=5r^2, divide by 5 first: A/5=r^2. The root applies to the whole fraction, r=√(A/5), not to the A only. A root of a product or a fraction covers everything that was on that side.
If the new subject is under a square root, square both sides once the root is isolated. From t=√(4d), square: t^2=4d, then d=dfract^24. Moving the 4 across without squaring leaves the root in place and does not make d the subject. At this grade take the positive square root when the letter stands for a length.
Method: 1. If the letter you want is squared, isolate that square and then square-root both sides. 2. From A=5r^2, divide by 5 first so that A/5=r^2. 3. The root covers the whole fraction: r=√(A/5), not the square root of A alone.
If the new subject is divided by something, multiply both sides by that something after you have isolated the fraction. s=dfracv^2-u^22a already has s as the subject. To make a the subject instead, multiply by 2a and then divide by 2s: 2as=v^2-u^2, so a=dfracv^2-u^22s. Dividing only one term of the top, as in dfracv^22s-u^2, drops the bracket that the subtraction needs.
Check with numbers. Let v=10, u=4 and a=3. Then v^2=u^2+2as gives 100=16+6s, so s=14. The rearranged formula s=dfracv^2-u^22a=84/6=14 agrees. One matching number is strong evidence. A mismatch means a term was left outside a division.
Choose the check numbers so the arithmetic stays kind. A subject that will be squared is easier to check with a whole number than with a fraction. If the rearranged formula has a square root, square your result and see whether you return to the original value. For r=√(A/5) with A=45, the root is 3, and 5× 3^2 is 45 again. That return journey is the check. If it fails, look at whether the root covered the whole fraction, and whether an earlier term was left outside the division. One clean number is enough. You do not need a second check unless the first one failed.
Method: 1. When the fraction is isolated, multiply both sides by its denominator. 2. Leave s=dfracv^2-u^22a as it is when s is the letter you want. 3. To make a the subject, multiply by 2a and then divide by 2s, giving a=dfracv^2-u^22s.
Many exam questions give a formula for a real situation, such as a taxi fare, a mobile phone bill or the perimeter of a rectangle, and then ask you to make one letter the subject. A typical layout is part (a) to substitute numbers, then part (b) to rearrange, then part (c) to use your new formula with a different number. Parts (b) and (c) link, so a correct rearrangement saves you work.
Make the new subject stand alone on one side with a coefficient of one. Keep the steps in the same order as solving an equation: undo adding or subtracting first, then undo multiplying or dividing. A formula like C = 2m + 3 becomes C - 3 = 2m and then m = (C - 3)/2. The answer is a formula, not a number, so it should still contain letters.
Method: 1. Write the letter you must isolate and circle it in the formula. 2. Undo addition or subtraction by doing the opposite to both sides. 3. Undo multiplication or division by doing the opposite to both sides. 4. Test the new formula with one pair of numbers from the original.
Tip: If part (c) gives you a number, put it into your rearranged formula rather than solving a fresh equation.
When the letter you want has a minus sign in front of it, as in y = 7 - 2x, move the number first to get y - 7 = -2x. Then divide both sides by -2 to find x = (y - 7)/(-2), which is the same as x = (7 - y)/2. Swapping the order in the top line removes the negative from the bottom, and both forms earn the marks.
A bracket can be handled in two ways. For y = 3(x - 5) you can divide both sides by 3 first to get y/3 = x - 5, then add 5. Or you can expand to y = 3x - 15 and carry on. Dividing first is quicker when the bracket multiplies the whole of one side. Both routes give x = y/3 + 5.
Method: 1. Move any term that is added or subtracted away from the letter you want. 2. If the letter has a minus sign, divide by the negative or swap the order on top. 3. For a bracket, divide by the number outside first, or expand it. 4. Substitute a simple number to check your answer.
In a harder question the letter you want appears in two places, for example y = (x + 5)/(x - 3). You cannot solve for x while it is on both sides of an equals sign, so you must gather it. First clear any fraction by multiplying both sides by the denominator. Then expand the bracket, so every x term is visible.
Next move all terms containing x to one side and everything else to the other. Factorise the x terms to pull x out as a single factor, so yx - x becomes x(y - 1). Finally divide by the bracket. This idea of collect, factorise and divide is the key skill, and it is worth a lot of the marks in a four mark question.
Method: 1. Multiply out any fraction so the equation has no denominator. 2. Expand brackets so each x term is separate. 3. Collect all x terms on one side and the rest on the other side. 4. Factorise out x, then divide by the bracket that remains.
Tip: The moment you see the same letter twice, think collect, factorise, divide.
A rearrangement is easy to check. Choose small whole numbers for the letters on the right of the original formula, work out the subject, then put your numbers into the new formula and see if you return to where you started. For d = 3h - 4 with h = 2 you get d = 2. Your rearranged formula must give h = 2 when d = 2.
Questions often show a pupil's method with one mistake and ask you to find it. Read the lines in order. The usual faults are adding when the formula says subtract, forgetting to divide every term by the same number, or dividing only part of one side. State the error in a short sentence, then write the line that should have been there, and finish the correct rearrangement.
Method: 1. Pick simple numbers and work out the original formula. 2. Put those results into your rearranged formula. 3. Check that you get your starting numbers back. 4. In an error question, find the first wrong line and correct from there.
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