Constraint System Examples in Math

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

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

Concept Recap

A collection of equations and inequalities that must ALL be satisfied simultaneously by the same set of variable values.

Multiple conditions at once: ' $x > 0$ AND $x + y = 10$ AND $y \leq 6$ .'

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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 constraint system requires the same values to satisfy every equation and inequality simultaneously.

Common stuck point: The procedure for constraint system is the easy part; the trap is satisfying some constraints but not all. Asking "Must the same values satisfy multiple conditions at the same time?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Must the same values satisfy multiple conditions at the same time?

Worked Examples

Example 1

medium

Find all values of

(x, y)

satisfying

x + y = 10

x \geq 0

, and

y \geq 0

Answer

All

(x, 10-x)

where

0 \leq x \leq 10

First step

From

x + y = 10

y = 10 - x

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Example 2

hard

A shop sells notebooks for \$3 and pens for \$1. You have \$12 and want at least 2 notebooks. How many pens can you buy?

Example 3

medium

Solve the system

x + y = 11

x - y = 3

Example 4

medium

Solve

3x + 4y = 20

and

x = 4

Example 5

medium

Solve

5x - 2y = 1

and

3x + 2y = 23

Example 6

hard

Maximize

x + 2y

subject to

x \le 4

y \le 5

x, y \ge 0

Example 7

challenge

A vending machine accepts only nickels and dimes. You insert

\$0.85

in exactly

11

coins. How many of each?

Practice Problems

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

Example 1

easy

List the integer solutions of

x + y = 5

with

x \geq 1

and

y \geq 1

Example 2

medium

(3, 4)

in the feasible region of

x + y \leq 8

and

x \geq 2

Example 3

easy

Does

x=4

satisfy both

x>0

and

x<10

Example 4

easy

Does

x=-2

satisfy

x>0

and

x<5

Example 5

easy

Which integer satisfies both

x\ge 2

and

x\le 2

Example 6

easy

x+y=5

and

x=2

, find

y

Example 7

easy

List the integers satisfying both

x>1

and

x<4

Example 8

easy

Does

(x,y)=(1,1)

satisfy

x+y=2

and

x-y=0

Example 9

easy

A length

x

satisfies

x<10

. What implicit constraint also applies?

Example 10

easy

x=6

in the solution set of the system

x\ge 5

and

x\le 8

Example 11

medium

Solve the system

x+y=10

x-y=4

Example 12

medium

Find all integers

x

with

x>0

x\le 6

, and

x

even.

Example 13

medium

Solve

2x+y=8

and

y=x-1

simultaneously.

Example 14

medium

A number

x

satisfies

x>0

x+y=10

, and

y\le 6

. Find the smallest allowed

x

Example 15

medium

Solve the system

x+2y=7

3x-2y=5

Example 16

medium

Determine whether the system

x+y=4

x+y=6

has a solution.

Example 17

challenge

Find all

(x,y)

with

x,y\ge0

integers and

2x+3y=12

Example 18

challenge

For what value of

k

does the system

x+y=4

2x+2y=k

have infinitely many solutions?

Example 19

challenge

Maximize

x+y

subject to

x\le 3

y\le 5

, and

x,y\ge 0

Example 20

medium

Solve

x-y=1

and

x+y=7

Example 21

medium

Find integers with

x\ge 1

x\le 5

, and

x

odd.

Example 22

medium

Solve

y=2x

and

x+y=9

Example 23

easy

List integers

x

satisfying

x \ge 3

and

x \le 6

Example 24

easy

Does

x = 0

satisfy

x > 0

and

x < 4

Example 25

easy

What is the only integer satisfying both

x \ge 5

and

x \le 5

Example 26

medium

Solve

2x + y = 9

and

x - y = 0

Example 27

medium

Find integer pairs with

x + y = 6

and

x, y \ge 0

Example 28

medium

Determine whether

x + 2y = 7

and

2x + 4y = 10

has a solution.

Example 29

medium

Find

(x, y)

with

x \ge 0

y \ge 0

, and

x + y = 7

that maximizes

x

Example 30

medium

List integer

(x, y)

with

x + y \le 3

x \ge 0

y \ge 0

Example 31

medium

Solve

x + y = 10

x - y = -2

Example 32

hard

Find all integers

(x, y)

with

x, y \ge 0

and

3x + 5y = 30

Example 33

hard

Find

k

so that

x + y = 5

and

2x + 2y = k

have infinitely many solutions.

Example 34

hard

Solve

x + y + z = 6

y = 2

z = 1

Example 35

challenge

How many lattice points satisfy

x + y \le 4

with

x, y \ge 0

integer?

Example 36

challenge

Solve

x + y + z = 9

x = y

z = 2y

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Related Concepts

Systems of Equations Inequalities Linear Programming

Background Knowledge

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

systems of equationsinequalities