Changing the Subject of a Formula

The subject of a formula

In the formula y = mx + c, y is the subject: it stands alone on one side, and the formula tells you how to work it out from x, m and c. Changing the subject means rearranging the formula so that a different letter stands alone.

A formula is an equation, so the rule is the one you use for solving equations: whatever you do to one side, do to the other. The difference is that the answer is an expression in the other letters, not a number.

What happens to x

Follow x through the formula y = mx + c. First x is multiplied by m, giving mx. Then c is added, giving y. To get x back from y, undo these steps in the reverse order: first undo the adding, then undo the multiplying.

Going forward, x is multiplied by m, and then c is added.

Going back, the last step is undone first: subtract c, and then divide by m.

Rearranging, one step at a time

Start with y = mx + c. Subtract c from both sides: y − c = mx.

Now divide both sides by m. The whole of the left side, y − c, is divided by m, not just the y: $\frac{{y - c}}{m} = x$. The bracket shows that the subtraction happens before the division.

It is usual to write the subject first, so turn the formula round: $x = \frac{{y - c}}{m}$. It is the same formula as y = mx + c, rearranged so that x stands alone.

Using the new formula

Take y = 3x + 5. Here m = 3 and c = 5, so $x = \frac{{y - 5}}{3}$. When y = 17, $x = \frac{{17 - 5}}{3} = \frac{12}{3} = 4$. Check in the original formula: 3 × 4 + 5 = 17.

The letter you want decides the steps

A formula can be rearranged for any of its letters. To make m the subject of y = mx + c, start the same way: subtract c from both sides, y − c = mx. This time m is multiplied by x, so divide both sides by x: $m = \frac{{y - c}}{x}$.

Another example: in v = u + at, make t the subject. Subtract u from both sides: v − u = at. Divide both sides by a: $t = \frac{{v - u}}{a}$.

Undo in the reverse order

The area of a circle is $A = \pi r^{2}$. The radius is squared first, then multiplied by $\pi$. To make r the subject, undo those steps last one first: divide both sides by $\pi$, so $r^{2} = \frac{A}{\pi}$, then take the square root, so $r = \sqrt{\frac{A}{\pi}}$. Taking the square root first undoes the steps in the wrong order and gives a different number.

Choose the order of the two undo steps. With $\div \pi$ first and then the square root, r comes back, and dragging the radius shows it comes back for every circle.

The usual mistakes

Dividing only part of a side. From y − c = mx, dividing by m gives $\frac{{y - c}}{m}$, not $\frac{y}{m} - c$. Every term on the side is divided.

Dividing by m first and forgetting the c. Dividing both sides of y = mx + c by m divides every term: $\frac{y}{m} = x + \frac{c}{m}$, not x + c. Subtracting c first, while it is still a separate term, avoids the fraction $\frac{c}{m}$.