Isolating Variable Examples in Math

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 xx until only xx remains: x=x = answer.

Read the full concept explanation β†’

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 yy in 2x+y=102x + y = 10.

Answer

y=10βˆ’2xy = 10 - 2x

First step

1
Subtract 2x2x from both sides: y=10βˆ’2xy = 10 - 2x.

Full solution

  1. 2
    Now yy is alone on one sideβ€”it is isolated.
  2. 3
    This expresses yy as a function of xx.
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=12bhA = \frac{1}{2}bh for hh.

Example 3

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

Example 4

medium
Solve for xx: 4xβˆ’9=234x - 9 = 23.

Practice Problems

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

Example 1

easy
Isolate xx in xβˆ’7=3x - 7 = 3.

Example 2

hard
Solve P=2l+2wP = 2l + 2w for ww.

Example 3

easy
Isolate xx: x+8=15x+8=15.

Example 4

easy
Isolate xx: 4x=204x=20.

Example 5

easy
Isolate xx: xβˆ’9=βˆ’2x-9=-2.

Example 6

easy
Isolate xx: x3=6\frac{x}{3}=6.

Example 7

easy
Isolate xx: 7+x=37+x=3.

Example 8

easy
Isolate xx: βˆ’x=5-x=5.

Example 9

easy
Isolate xx: 2x=72x=7.

Example 10

easy
Isolate xx: x+12=2x+\frac{1}{2}=2.

Example 11

medium
Isolate xx: 3x+7=193x+7=19.

Example 12

medium
Isolate xx: x2βˆ’5=1\frac{x}{2}-5=1.

Example 13

medium
Isolate xx: 5xβˆ’3=2x+95x-3=2x+9.

Example 14

medium
Isolate xx: 2(x+3)=142(x+3)=14.

Example 15

medium
Solve for rr: C=2Ο€rC=2\pi r.

Example 16

medium
Solve for hh: A=12bhA=\frac{1}{2}bh.

Example 17

medium
Isolate xx: 2x+13=5\frac{2x+1}{3}=5.

Example 18

challenge
Solve for xx: ax+b=cax+b=c (with a≠0a\ne0), in terms of a,b,ca,b,c.

Example 19

challenge
Solve for xx: 1x+12=34\frac{1}{x}+\frac{1}{2}=\frac{3}{4}.

Example 20

challenge
Solve for tt: s=ut+12at2s=ut+\frac{1}{2}at^2 is hard; instead solve v=u+atv=u+at for tt.

Example 21

medium
Isolate xx: x+42=6\frac{x+4}{2}=6.

Example 22

medium
Solve for bb: A=a+b2A=\frac{a+b}{2}.

Example 23

easy
Solve for xx: x+12=5x + 12 = 5.

Example 24

easy
Solve for yy: yβˆ’11=4y - 11 = 4.

Example 25

easy
Solve for xx: 6x=426x = 42.

Example 26

easy
Solve for xx: x5=βˆ’2\frac{x}{5} = -2.

Example 27

easy
Solve for xx: 2x3=8\frac{2x}{3} = 8.

Example 28

medium
Solve for xx: 7βˆ’2x=17 - 2x = 1.

Example 29

medium
Solve for xx: 3(xβˆ’4)=183(x - 4) = 18.

Example 30

medium
Solve for xx: 5(2x+1)βˆ’4=215(2x + 1) - 4 = 21.

Example 31

medium
Solve for xx: 4x+7=2x+194x + 7 = 2x + 19.

Example 32

medium
Solve for hh: V=Ο€r2hV = \pi r^2 h.

Example 33

medium
Solve for CC: F=95C+32F = \frac{9}{5}C + 32.

Example 34

medium
Solve for xx: xβˆ’34=x+16\frac{x - 3}{4} = \frac{x + 1}{6}.

Example 35

hard
Solve for xx: 2(3xβˆ’1)βˆ’4(x+5)=62(3x - 1) - 4(x + 5) = 6.

Example 36

hard
Solve for xx: 2xβˆ’13=16\frac{2}{x} - \frac{1}{3} = \frac{1}{6} (with xβ‰ 0x \neq 0).

Example 37

hard
Solve for rr: A=P(1+rt)A = P(1 + rt) (simple interest).

Example 38

hard
Solve for xx: 2x+5=7\sqrt{2x + 5} = 7.

Example 39

hard
Solve for nn: S=n(n+1)2S = \frac{n(n+1)}{2} in terms of SS (positive solution).

Example 40

hard
Solve for xx: x+2xβˆ’1=3\frac{x + 2}{x - 1} = 3 (with xβ‰ 1x \neq 1).

Example 41

hard
Solve for xx in terms of a,ba, b: xβˆ’ab=x+ba\frac{x - a}{b} = \frac{x + b}{a} (a,bβ‰ 0a, b \neq 0, aβ‰ ba \neq b).

Example 42

challenge
Solve for xx: 2x+1=162^{x+1} = 16.

Example 43

challenge
Solve for xx: log⁑2(xβˆ’1)+log⁑2(x+1)=3\log_2(x - 1) + \log_2(x + 1) = 3.
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Background Knowledge

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

inverse operationsequations