Substitution Examples in Math

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

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

Concept 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.

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: Substitution replaces every occurrence of a variable with an equal value or expression.

Common stuck point: The procedure for substitution is the easy part; the trap is dropping parentheses around the substituted expression. Asking "Am I replacing a variable with an EQUAL expression everywhere it appears?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I replacing a variable with an EQUAL expression everywhere it appears?

Worked Examples

Example 1

easy

y = 3x + 1

and

x = 2

, find

y

Answer

y = 7

First step

Replace

x

with 2 in the expression:

y = 3(2) + 1

Full solution

2
Compute: $y = 6 + 1 = 7$ .
3
Substitution replaces a variable with its known value.

Substitution means replacing a variable with an equivalent value or expression. Here we replace

x

with 2 to find the corresponding

y

Example 2

medium

y = x + 3

and

2x + y = 9

, use substitution to solve for

x

Example 3

easy

Given

f(x) = 5 - x

, evaluate

f(-2)

by substitution.

Example 4

medium

y = x - 4

and

2x + 3y = 7

, use substitution to find

x

Example 5

medium

Let

u = x^2

. Rewrite

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

in terms of

u

and factor.

Practice Problems

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

Example 1

easy

Evaluate

a + 2b

when

a = 5

and

b = -1

Example 2

medium

u = 2v - 1

and

3u + v = 11

, find

v

Example 3

easy

y=2x

and

x=4

, find

y

Example 4

easy

Given

a=5

, substitute into

a+3

Example 5

easy

y=x+1

, substitute to express

2y

in terms of

x

Example 6

easy

x=3

and

z=x^2

, find

z

Example 7

easy

b=-2

, substitute into

4-b

Example 8

easy

u=3v

and

v=2

, find

u

Example 9

easy

t=x-1

, replace

x

with

5

to find

t

Example 10

easy

Given

p=2

and

q=p+4

, then

r=q-1

. Find

r

Example 11

medium

y=2x

, substitute to eliminate

y

from

3y+x=14

, then solve for

x

Example 12

medium

Given

a=b+2

and

a+b=10

, substitute to find

b

Example 13

medium

y=x+1

, substitute into

y^2

and expand.

Example 14

medium

Given

f(x)=3x-2

, find

f(x+1)

by substitution.

Example 15

medium

s=t^2

and

t=x-3

, express

s

in terms of

x

Example 16

medium

Given

2x+y=7

and

y=3

, substitute to find

x

Example 17

challenge

Given

x+y=6

and

xy=8

, use substitution to find both values of

x

Example 18

challenge

g(x)=x^2

and

h(x)=2x-1

, find

g(h(x))

and evaluate it at

x=2

Example 19

challenge

ax+b=c

, substitute

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

and verify it satisfies the equation (with

a\ne0

Example 20

medium

y=x-2

, substitute into

x+y

and simplify.

Example 21

medium

Given

a=2b

and

b=t+1

, express

a

in terms of

t

Example 22

medium

3x-y=7

and

x=4

, substitute to find

y

Example 23

easy

m = 4n

and

n = 7

, find

m

Example 24

easy

Evaluate

3a - 2b

when

a = 4

and

b = 5

Example 25

easy

h = 2k + 3

and

k = 5

, find

h

Example 26

easy

a = \frac{1}{2}

and

b = 4

, find

ab

Example 27

easy

Evaluate

x^2 - 4

x = 6

Example 28

medium

Given

f(x) = x^2 + 1

, find

f(2a)

as an expression in

a

Example 29

medium

p = 2q + 1

and

3p - q = 13

, find

q

Example 30

medium

u = a + b

and

v = a - b

, express

u^2 - v^2

in terms of

a

and

b

Example 31

medium

Given

g(x) = 4x + 7

, find

g(x - 3)

as a simplified expression.

Example 32

medium

Given

a = 3

b = -1

, find

a^2 - 2ab + b^2

Example 33

medium

m = n + 4

and

n = 2t

, express

m

in terms of

t

Example 34

medium

Given

f(x) = 2x + 5

, find

x

such that

f(x) = 17

Example 35

hard

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

, find

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

Example 36

hard

Given

f(x) = x^2 - 2x

and

g(x) = x + 3

, find

f(g(2))

Example 37

hard

Solve the system

y = x^2

and

y = 2x + 3

by substitution.

Example 38

hard

Use the substitution

u = x + 1

to simplify

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

and factor the result in terms of

u

Example 39

hard

Given the system

3x - y = 2

and

y = x^2 - 2

, find all real

x

Example 40

challenge

x + y = 4

and

x^3 + y^3 = 28

, find

xy

Example 41

challenge

Let

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

. Compute

f(f(x))

and simplify.

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Related Concepts

Equations Equal Systems of Equations

Background Knowledge

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

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