Systems of Equations Examples in Math

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

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

Concept Recap

Two or more equations sharing the same variables, where the solution must satisfy all equations simultaneously.

Where two lines cross—the point that satisfies both equations.

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: A system solution is the place where all constraints agree.

Common stuck point: The procedure for systems of equations is the easy part; the trap is solving only one equation and ignoring the other. Asking "Does the answer need to make every equation true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the answer need to make every equation true?

Worked Examples

Example 1

easy

Solve the system:

x + y = 10

and

x - y = 4

Answer

x = 7, \quad y = 3

First step

Add the two equations to eliminate

y

(x+y)+(x-y) = 10+4

, giving

2x = 14

Full solution

2
Solve for $x$ : $x = 7$ .
3
Substitute back into $x + y = 10$ : $7 + y = 10$ , so $y = 3$ .
4
Check in second equation: $7 - 3 = 4$ ✓

The elimination method adds or subtracts equations to remove one variable. This works well when coefficients of one variable are equal (or opposites).

Example 2

medium

Solve the system:

y = 2x + 1

and

3x + y = 11

Example 3

hard

Solve the system:

2x + 3y = 12

and

x - y = 1

Example 4

easy

Solve:

x + 2y = 8

x - y = 2

Example 5

medium

Solve:

\frac{x}{2} + y = 5

x - y = 1

Example 6

medium

At a fair,

3

child tickets and

2

adult tickets cost $31.

5

child and

4

adult tickets cost $57. Find each price.

Example 7

hard

Mixing problem: a chemist combines a

20\%

acid and a

50\%

acid solution to make

30

L of

40\%

acid. How many liters of each?

Example 8

hard

A boat travels

30

miles downstream in

2

hours and returns upstream in

3

hours. Find the boat's speed in still water and the current's speed.

Practice Problems

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

Example 1

easy

Solve:

x + y = 8

and

x = 3

Example 2

hard

Solve:

2x + 3y = 12

and

4x - y = 5

Example 3

easy

Solve by substitution:

y = 2x

x + y = 6

Example 4

easy

(1, 2)

a solution of

x + y = 3

and

x - y = -1

Example 5

easy

Solve by elimination:

x + y = 5

x - y = 1

Example 6

easy

Two lines have the same slope but different intercepts. How many solutions?

Example 7

easy

If two equations describe the SAME line, how many solutions?

Example 8

easy

Solve:

x = 4

y = x - 1

Example 9

easy

In a 2-variable system, the solution is described as what kind of object?

Example 10

easy

Make coefficients match: multiply

x + 2y = 4

so the

x

-coefficient is

3

Example 11

medium

Solve:

2x + 3y = 12

x = y

Example 12

medium

Solve by elimination:

3x + 2y = 16

x + 2y = 8

Example 13

medium

Solve:

x + y = 10

2x - y = 2

Example 14

medium

Multiply and eliminate:

2x + y = 7

x + 3y = 7

Example 15

medium

How many solutions does

x + y = 4

2x + 2y = 8

have?

Example 16

medium

How many solutions does

x + y = 3

x + y = 5

have?

Example 17

medium

Tickets: adult $8, child $5, total $81 for

12

tickets. Set up the system.

Example 18

challenge

Solve the ticket system

a + c = 12

8a + 5c = 81

Example 19

challenge

Three equations, two unknowns, all consistent. What does the third equation likely add?

Example 20

challenge

For what

k

does

x + y = 2

2x + 2y = k

have infinitely many solutions?

Example 21

medium

Solve:

y = x + 1

y = 2x - 3

Example 22

medium

Solve:

2x + y = 5

y = x + 1

y = -x + 5

Example 33

medium

For what value of

b

does the system

3x + y = 5

6x + 2y = b

have infinitely many solutions?

Example 34

hard

Solve:

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

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

Example 35

hard

Solve:

x + y + z = 6

x - y + z = 2

2x + y - z = 1

Example 36

hard

For what

k

does the system

kx + 2y = 6

3x - y = 4

have no solution?

Example 37

hard

Find

(x, y)

3x + 4y = 22

5x - 2y = -7

Example 38

challenge

Three numbers sum to

30

. The first is half the second. The third is twice the first. Find the three numbers.

Example 39

challenge

For what

k

does the system

2x + ky = 4

kx + 8y = 8

have infinitely many solutions?

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Related Concepts

Linear Functions Solving Linear Equations Linear Programming

Background Knowledge

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

linear functionssolving linear equations