Practice Substitution in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Replacing every occurrence of a variable or sub-expression with an equivalent value or expression throughout a problem.

If $y = 2x$ , you can write $2x$ everywhere you see $y$ —they're the same.

Showing a random 20 of 50 problems.

Example 1

hard

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

, find

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

Example 2

challenge

Let

f(x) = \frac{x}{1 - x}

. Compute

f(f(x))

and simplify.

Example 3

easy

t=x-1

, replace

x

with

5

to find

t

Example 4

hard

a + b = 5

and

a^2 + b^2 = 13

, find

ab

Example 5

medium

y = x - 4

and

2x + 3y = 7

, use substitution to find

x

Example 6

medium

x + y = 5

and

y = 2x - 1

, find

x

Example 7

easy

a = \frac{1}{2}

and

b = 4

, find

ab

Example 8

medium

3x-y=7

and

x=4

, substitute to find

y

Example 9

easy

Evaluate

x^2 - 4

x = 6

Example 10

easy

x=3

and

z=x^2

, find

z

Example 11

challenge

g(x)=x^2

and

h(x)=2x-1

, find

g(h(x))

and evaluate it at

x=2

Example 12

medium

p = 2q + 1

and

3p - q = 13

, find

q

Example 13

medium

Given

a=2b

and

b=t+1

, express

a

in terms of

t

Example 14

medium

y=x+1

, substitute into

y^2

and expand.

Example 15

hard

Use the substitution

u = x + 1

to simplify

(x + 1)^3 - 4(x + 1)

and factor the result in terms of

u

Example 16

medium

Let

u = x^2

. Rewrite

x^4 - 5x^2 + 6 = 0

in terms of

u

and factor.

Example 17

challenge

ax+b=c

, substitute

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

and verify it satisfies the equation (with

a\ne0

Example 18

medium

Given

a = 3

b = -1

, find

a^2 - 2ab + b^2

Example 19

medium

m = n + 4

and

n = 2t

, express

m

in terms of

t

Example 20

medium

Given

f(x) = x^2 + 1

, find

f(2a)

as an expression in

a

← Back to Substitution Worked Examples Formula Explained

Related Concepts

Equations Equal Systems of Equations