Isolating Variable Examples: 56 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Isolating Variable.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Rearranging an equation by applying inverse operations until the variable stands alone on one side.

Peel away everything around $x$ until only $x$ remains: $x =$ answer.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Isolating a variable undoes each operation around it in reverse order using inverses.

Common stuck point: The procedure for isolating variable is the easy part; the trap is undoing multiplication before addition. Asking "Am I peeling operations off the variable until it stands alone on one side?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I peeling operations off the variable until it stands alone on one side?

Worked Examples

Example 1

easy

Isolate

y

2x + y = 10

Answer

y = 10 - 2 x

First step

Subtract

2x

from both sides:

y = 10 - 2x

Full solution

2
Now $y$ is alone on one side—it is isolated.
3
This expresses $y$ as a function of $x$ .

Isolating a variable means getting it alone on one side of the equation. This is done by performing inverse operations on both sides.

Example 2

medium

Solve

A = \frac{1}{2}bh

for

h

Example 3

medium

Solve for y: 3(2y - 4) + 5 = 2y + 9

Example 4

medium

Solve for

x

4x - 9 = 23

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Isolate

x

x - 7 = 3

Example 2

hard

Solve

P = 2l + 2w

x

ax+b=c

(with

a\ne0

), in terms of

a,b,c

Example 19

challenge

Solve for

x

\frac{1}{x}+\frac{1}{2}=\frac{3}{4}

Example 20

challenge

Solve for

t

s=ut+\frac{1}{2}at^2

is hard; instead solve

v=u+at

x

\frac{2}{x} - \frac{1}{3} = \frac{1}{6}

(with

x \neq 0

Example 37

hard

Solve for

r

A = P(1 + rt)

(simple interest).

Example 38

hard

Solve for

x

\sqrt{2x + 5} = 7

Example 39

hard

Solve for

n

S = \frac{n(n+1)}{2}

in terms of

S

(positive solution).

Example 40

hard

Solve for

x

\frac{x + 2}{x - 1} = 3

(with

x \neq 1

Example 41

hard

Solve for

x

in terms of

a, b

\frac{x - a}{b} = \frac{x + b}{a}

(

a, b \neq 0

a \neq b

Example 42

challenge

Solve for

x

2^{x+1} = 16

Example 43

challenge

Solve for

x

\log_2(x - 1) + \log_2(x + 1) = 3

Example 44

easy

Fill in the blank so that the solution is

x = 4

3x + \_\_ = 17

Example 45

easy

Fill in the blank so that the solution is

x = 6

\frac{x}{2} + \_\_ = 10

Example 46

medium

Fill in the blank coefficient so that the solution is

x = 5

\_\_ \cdot x - 4 = 26

Example 47

medium

Jordan solves

2x + 6 = 18

like this: "Divide by 2:

x + 6 = 9

. Subtract 6:

x = 3

." Find the mistake and enter the correct value of

x

Example 48

medium

Priya solves

\frac{x}{3} - 2 = 4

like this: "Multiply both sides by 3:

x - 2 = 12

. Add 2:

x = 14

." Find the mistake and enter the correct value of

x

Example 49

easy

Solve for

x

3x + 6 = 21

Example 50

medium

Solve for

x

3(x + 6) = 21

Example 51

easy

A number machine takes a number

x

, multiplies it by 4, then adds 9, and the display shows 37. What number went into the machine?

Example 52

medium

A taxi ride costs a

\$3

booking fee plus

\$2

for every mile. The whole ride costs

\$19

. How many miles was the ride?

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Related Concepts

Inverse Operations Equations Solving Linear Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

inverse operationsequations