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Rearrange formulas to isolate a variable

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.

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

to make yy the subject.

  1. Expand the brackets:

18=3x+2y−818=3x+2y-8

  1. Move the terms that do not contain yy to the other side.
    Add 88 and subtract 3x3x from both sides:

18+8−3x=2y18+8-3x=2y

Simplify:

26−3x=2y26-3x=2y

  1. Divide both sides by the coefficient of yy, which is 22:

26−3x2=y\frac{26-3x}{2}=y

Therefore,

y=26−3x2\boxed{y=\frac{26-3x}{2}}

An equivalent form is

y=13−3x2\boxed{y=13-\frac{3x}{2}}

Check the result

Choose x=2x=2. The original formula gives

18=3(2)+2(y−4)18=3(2)+2(y-4) 18=6+2(y−4)18=6+2(y-4)

so y=10y=10.

Using the rearranged formula:

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

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

Learn by doing: Rearrange formulas to isolate a variable

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    Linear Equation - One Variable, Three Terms

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