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Exercises on systems of equations substitution and elimination

Solve systems of two equations with two unknowns by substitution, equating, and elimination. Theory and step-by-step solved exercises, with practice instantly corrected by AI.

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Systems of equations: quick theory

A system of equations is two (or more) equations that must be true at the same time. Solving it means finding the values of x and y that satisfy both.

Three methods: substitution (solving for one variable and substituting it into the other equation), equating (solving for the same variable in both and setting them equal), and elimination (adding or subtracting the equations to eliminate a variable).

Step-by-step solved exercises on systems of equations

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1Solve by substitution: x + y = 5 ; x − y = 1.

Step-by-step solution

  1. From the first one, we solve for x = 5 − y.
  2. We substitute into the second: (5 − y) − y = 1 → 5 − 2y = 1.
  3. We isolate y: −2y = −4 → y = 2.
  4. Going back: x = 5 − 2 = 3.
Solution: x = 3, y = 2.

Exercise 2Solve by elimination: x + y = 10 ; 2x − y = 2.

Step-by-step solution

  1. We add the two equations (the y cancels out): 3x = 12.
  2. We isolate x: x = 4.
  3. We substitute into x + y = 10: 4 + y = 10 → y = 6.
Solution: x = 4, y = 6.

Exercise 3Solve: 3x + 2y = 16 ; x + 2y = 8.

Step-by-step solution

  1. We subtract the second from the first (the 2y cancels out): 2x = 8.
  2. We isolate x: x = 4.
  3. We substitute into x + 2y = 8: 4 + 2y = 8 → 2y = 4 → y = 2.
Solution: x = 4, y = 2.

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This topic is studied in:

2nd year of ESO3rd year of ESO4th year of ESO

Frequently asked questions about systems of equations

Which method is best for solving a system of equations?

It depends on the system. Substitution works well when a variable is easy to isolate; elimination is convenient when the coefficients of a variable are equal or opposite. All three give the same result.

When does a system have no solution?

When the two equations represent parallel lines (they are inconsistent). If they represent the same line, there are infinitely many solutions.

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