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What Is the Systems of Equations?(With Examples)

A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are true simultaneously. For linear systems, this is where the lines intersect on a graph.

Grades 8-9 (Algebra)GCSE (Year 10-11)Årskurs 9Klasse 8-9

📖Definition

A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are true simultaneously. For linear systems, this is where the lines intersect on a graph.

📐Formula

System: { ax + by = c, dx + ey = f }

A system of two linear equations can have one solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same line).

📝Step-by-Step Guide

1

Substitution Method

Solve one equation for one variable, then substitute into the other equation.

2

Elimination Method

Add or subtract equations to eliminate one variable.

Multiply equations if needed to match coefficients
3

Solve for First Variable

Use algebra to find the value of one variable.

4

Find the Second Variable

Substitute back into either original equation to find the other variable. Check in both equations.

⚠️Common Mistakes to Avoid

  • Arithmetic errors when eliminating variables
  • Substituting into the wrong equation
  • Forgetting to find both variables
  • Not checking the solution in both equations
  • Misidentifying parallel lines (no solution)

✏️Practice Problems

Easy

Solve: x + y = 10, x - y = 4

Answer: x = 7, y = 3

Medium

Solve: 2x + 3y = 12, x - y = 1

Answer: x = 3, y = 2

Hard

Solve: 3x - 2y = 8, 2x + 5y = -1

Answer: x = 2, y = -1

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Worked Examples

System of Equations Examplemedium

🔗Related Topics

Linear EquationsGraphingSubstitutionMatrices

Curriculum Alignment

CommonCore (8.EE.C.8)GCSEKMK

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