Help.ai Learn · Skill guide

Systems of equations

Solve two linear equations together using their shared variables. Connect elimination or substitution to the intersection of the corresponding relationships.

Read the guide See a worked example

What you’ll learn

Explain why a system’s solution must satisfy every equation.
Use elimination or substitution for a simple linear system.
Recognize that a system may have one, no, or many solutions.

One answer must fit all the relationships

A pair that satisfies only one equation is not a solution to the system. For two linear equations in two variables, graphing can show the shared point, parallel lines, or the same line.

Remove one unknown without losing the relationship

Elimination combines equations to cancel a variable; substitution replaces a variable with an equivalent expression from another equation. Choose the method that keeps the arithmetic clear.

A mistake to avoid

When adding or scaling equations, include every term on both sides. If the calculation produces a contradiction or identity, interpret it instead of forcing a single numerical answer.

Worked example

Solve x + y = 7 and x − y = 1

Add the equations to cancel y: 2x = 8, so x = 4. Substitute into x + y = 7 to find y = 3. Check both: 4 + 3 = 7 and 4 − 3 = 1.

Learning paths that include this skill

Choose the route that fits your goal. The same skill can be useful in more than one subject or stage of learning.

Secondary / high-school algebraHigh-school algebraWork through the relationships behind secondary algebra. Reuse number skills, connect equations to graphs, and build toward systems and quadratics.Signed numbers · Equivalent fractions · Decimals · Linear equations · +4 more Preparing for calculusPrecalculus foundationsReview equation, graph, and angle foundations before calculus. Identify the prerequisite that needs attention and connect formal notation to its meaning.Linear equations · Systems of equations · Quadratic equations · Graphs & coordinates · +5 more Introductory linear algebraLinear algebra foundationsBuild from equations and coordinates to vectors and matrices. Explain what each component represents and how a matrix-vector calculation combines quantities.Signed numbers · Graphs & coordinates · Linear equations · Systems of equations · +2 more

Graphs & coordinatesRead a graph as a relationship between quantities. Check axes and scale, connect points to an equation, and interpret the change between two points.VectorsRepresent several components as one vector and explain addition or scaling. Connect component calculations to direction, size, and a coordinate picture.MatricesRead a matrix by its rows and columns and calculate a matrix-vector product. Connect the result to coefficients, linear combinations, or a transformation.

A question worth talking through.

Explore the AI Tutor we’re developing for clear explanations, useful follow-up questions, and guided practice.

Explore AI Tutor