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.'

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: Every constraint must be satisfied at once — the solution lies in the intersection of all feasible regions.

Common stuck point: Adding more constraints reduces the feasible region — too many constraints may leave no solution at all.

Sense of Study hint: List every condition on separate lines, then check your answer against each one individually.

Worked Examples

Example 1

medium
Find all values of (x, y) satisfying x + y = 10, x \geq 0, and y \geq 0.

Solution

  1. 1
    From x + y = 10: y = 10 - x.
  2. 2
    Apply x \geq 0: this gives a lower bound on x.
  3. 3
    Apply y \geq 0: 10 - x \geq 0, so x \leq 10.
  4. 4
    The feasible set is all (x, 10-x) with 0 \leq x \leq 10.

Answer

All (x, 10-x) where 0 \leq x \leq 10.
A constraint system combines equations and inequalities that must all be satisfied simultaneously. The solution is the set of points meeting every constraint.

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?

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
Is (3, 4) in the feasible region of x + y \leq 8 and x \geq 2?
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Background Knowledge

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

systems of equationsinequalities