Systems of Equations
Solve systems using various methods
lessons (6)
Introduction to Systems of Equations
Learn what systems of equations are and discover how two equations can work together to find a unique solution.
Systems of Equations (Substitution)
Learn to solve systems of two equations using the substitution method.
Solving Systems by Graphing
Learn how to solve systems of linear equations by graphing both lines and finding their intersection point.
Substitution Method
Learn to solve systems of equations by substituting one equation into another.
Elimination Method
Learn to solve systems of equations by eliminating one variable through addition or subtraction.
Systems of Equations Word Problems
Learn to translate real-world situations into systems of equations and solve them step by step.
Systems of equations allow us to solve problems with multiple unknowns simultaneously. When one equation is not enough to find a unique solution, we use two or more equations together. These systems appear everywhere: from balancing chemical equations to optimizing business decisions to analyzing traffic patterns.
In this topic, you will master three methods for solving systems: graphing, substitution, and elimination. Each method has its strengths—graphing provides visual insight, substitution works well when one variable is isolated, and elimination is efficient for systems with convenient coefficients. Understanding when to use each method is a key problem-solving skill.
Our lessons progress from two-variable systems to applications including mixture problems, motion problems, and break-even analysis. You will learn to interpret solutions geometrically (intersecting lines, parallel lines, same line) and apply systems thinking to real-world scenarios.
What Students Will Learn
- Solve systems by graphing and interpret the solution
- Apply the substitution method effectively
- Use the elimination method with multiplication
- Determine if a system has one, none, or infinitely many solutions
- Set up systems from word problems
- Solve mixture and motion problems using systems
- Extend to systems with three variables
Frequently Asked Questions
What does the solution to a system represent graphically?
The solution is the point where the lines intersect. One solution means the lines cross at exactly one point. No solution means parallel lines (never intersect). Infinitely many solutions means the same line (all points in common).
When should I use substitution vs. elimination?
Use substitution when one equation easily solves for a variable (like y = 2x + 3). Use elimination when coefficients are set up for easy cancellation, or can be made so with multiplication.
How do I set up a system from a word problem?
Identify the unknowns and assign variables. Then translate each condition or relationship in the problem into an equation. You need as many equations as unknowns to solve the system.