Skill: Rearrange formulas to isolate a variable

Explanation and Free Practice Resources

Rearranging a formula means treating all other quantities as constants and applying equivalent operations—expanding, combining like terms, collecting the target variable, and dividing by a nonzero coefficient—to express that variable alone. The learner preserves equality, reverses an inequality when multiplying or dividing by a negative quantity, observes restrictions created by denominators, and verifies equivalent forms by substitution. The scope is linear formulas with numerical or fixed symbolic coefficients, not nonlinear rearrangements involving variables in exponents, products, or denominators.

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Detailed Explanation: Rearrange formulas to isolate a variable

To isolate a variable, treat every other variable as a constant. Use inverse operations on both sides of the equation so that equality is preserved.

Example

Rearrange the formula

18=3x+2(y-4)

to make $y$ the subject.

Expand the brackets:

18=3x+2y-8

Move the terms that do not contain $y$ to the other side.
Add $8$ and subtract $3x$ from both sides:

18+8-3x=2y

Simplify:

26-3x=2y

Divide both sides by the coefficient of $y$ , which is $2$ :

\frac{26-3x}{2}=y

Therefore,

\boxed{y=\frac{26-3x}{2}}

An equivalent form is

\boxed{y=13-\frac{3x}{2}}

Check the result

Choose $x=2$ . The original formula gives

18=3(2)+2(y-4)

18=6+2(y-4)

so $y=10$ .

Using the rearranged formula:

y=13-\frac{3(2)}{2}=13-3=10

Both forms give the same value, so the rearrangement is correct.

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