8th Grade Math

Algebra foundations, geometry, and mathematical modeling for eighth grade students.

Linear Equations and ExpressionsFunctions and GraphsIntro to Geometry: Angles and Shapes
Systems of EquationsPythagorean TheoremVolume and Surface Area of 3D Shapes
Budgeting and Saving with AlgebraGeometry in Construction and DesignModeling and Interpreting Data
Breaking Down Word ProblemsChecking Your Work and Estimating
Linear Equations and...Functions and GraphsIntro to Geometry: A...Systems of EquationsPythagorean TheoremVolume and Surface A...Budgeting and Saving...Geometry in Construc...Modeling and Interpr...Breaking Down Word P...Checking Your Work a...
Advanced Topics

Systems of Equations

What Is a System of Equations?

A system of equations is a set of two or more equations with the same variables. You're looking for values that work in all equations at once.

How to Solve

  • Substitution: Solve one equation for a variable, then plug it into the other.
  • Elimination: Add or subtract equations to eliminate a variable.

When Do We Use Them?

Systems help answer questions like: If you buy 3 apples and 2 bananas for $5, and 2 apples and 1 banana for $3, how much does each fruit cost?

Checking Your Solution

Always check your answers in both original equations!

Key Formula

\[\begin{cases} x + y = 10 \ 2x - y = 5 \end{cases}\]

Examples

  • Solve \(x + y = 6\) and \(x - y = 2\): Add to get \(2x = 8\), so \(x = 4\), then \(y = 2\).

  • If \(2a + b = 7\) and \(a - b = 1\), add and subtract to solve for \(a\) and \(b\).

In a Nutshell

Systems of equations help find solutions to problems with more than one equation.

Key Terms

System of Equations
A set of equations sharing variables.
Substitution
Replacing a variable with an equivalent expression.
Elimination
Adding or subtracting equations to remove a variable.
Next
Pythagorean Theorem