Algebraic Manipulation Examples: 47 Problems with Answers

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

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

Concept Recap

Algebraic manipulation is the process of rewriting expressions or equations into equivalent forms to reveal structure or solve for unknowns.

It is like rearranging a sentence without changing its meaning.

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: Algebraic manipulation turns an expression or equation into an equivalent one to expose structure or isolate an unknown, without ever changing which values make it true.

Common stuck point: The procedure for algebraic manipulation is the easy part; the trap is changing only one side of an equation. Asking "Does every step keep exactly the same set of true values, just written differently?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every step keep exactly the same set of true values, just written differently?

Worked Examples

Example 1

easy

Simplify

2(x + 3) - x

Answer

x + 6

First step

Step 1: Distribute:

2x + 6 - x

Full solution

2
Step 2: Combine like terms: $x + 6$ .
3
Check: At $x = 1$ : $2(4) - 1 = 7$ and $1 + 6 = 7$ ✓

Algebraic manipulation is the process of rewriting expressions using distribution and combining like terms to reach a simpler or more useful equivalent form.

Example 2

medium

Solve for

y

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

Example 3

medium

Solve

4x - 7 = 2x + 5

Example 4

medium

Expand and simplify

(x - 3)(x + 5)

Example 5

hard

Solve

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

for

x

Example 6

hard

Solve

\sqrt{x + 7} = x - 5

for real

x

Practice Problems

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

Example 1

easy

Simplify

5x - 3 + 2x + 7

Example 2

medium

Rearrange

V = \frac{1}{3}\pi r^2 h

to solve for

h

Example 3

easy

Simplify

3x + 2x

Example 4

easy

Expand

2(x + 5)

Example 5

easy

Solve

x + 7 = 12

Example 6

easy

Solve

3x = 21

Example 7

easy

Factor

6x + 9

Example 8

easy

Simplify

\dfrac{4x}{2}

Example 9

easy

Combine

5a - 3a + a

Example 10

easy

Expand

-(x - 4)

Example 11

medium

Solve

2x + 3 = 11

Example 12

medium

Solve

5x - 2 = 3x + 8

Example 13

medium

Simplify

\dfrac{x+3}{x}

as far as possible.

Example 14

medium

Solve for

y

3x + 2y = 12

Example 15

medium

Expand and simplify

(x+2)(x+3)

Example 16

medium

Solve

\dfrac{x}{4} + 1 = 3

Example 17

medium

Factor

x^2 + 7x + 12

Example 18

medium

Solve

2(x - 1) = x + 4

Example 19

challenge

Solve for

x

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

Example 20

challenge

Rewrite

x^2 + 6x + 5

by completing the square.

Example 21

challenge

x + \dfrac{1}{x} = 4

, find

x^2 + \dfrac{1}{x^2}

Example 22

medium

Solve

\dfrac{2x - 1}{3} = 5

Example 23

easy

Simplify

4a + 2 - a + 5

Example 24

easy

Expand

3(2x - 4)

Example 25

easy

Factor

10x - 15

Example 26

easy

Expand

-3(x + 2)

Example 27

easy

Combine

6y + 4 - 2y - 1

Example 28

medium

Solve

3(x - 2) = 2x + 5

Example 29

medium

Rearrange

A = \dfrac{1}{2}bh

to solve for

b

Example 30

medium

Simplify

\dfrac{6x + 9}{3}

Example 31

medium

Solve

\dfrac{x + 4}{2} = 7

Example 32

medium

Solve for

r

C = 2\pi r

Example 33

medium

Factor

x^2 - 5x - 24

Example 34

medium

Simplify

\dfrac{x^2 - 9}{x - 3}

for

x \ne 3

Example 35

hard

Solve

\dfrac{1}{x - 1} + \dfrac{1}{x + 1} = \dfrac{1}{2}

for the larger real

x

Example 36

hard

Rearrange

\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}

to solve for

v

Example 37

hard

Solve

|2x - 5| = 9

Example 38

hard

Simplify

\dfrac{x^2 - x - 6}{x^2 - 9}

for

x \ne \pm 3

Example 39

hard

Solve the system

2x + 3y = 12

and

x - y = 1

Example 40

challenge

a + b = 7

and

a^2 + b^2 = 29

, find

ab

Example 41

challenge

Simplify

\left( \dfrac{1}{x} - \dfrac{1}{y} \right) \div \left( \dfrac{x - y}{xy} \right)

for nonzero

x \ne y

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Related Concepts

Expressions Equivalence Transformation Balance Principle

Background Knowledge

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

expressionsequivalence transformationbalance principle