Why this chapter matters
Systems answer comparison questions such as when two plans cost the same or which combination of quantities meets two constraints.
What you will learn
- Solve two-variable systems by three standard methods.
- Recognize systems with one, none, or infinitely many solutions.
- Translate a two-condition word problem into a system.
Understand the core ideas
A system of two linear equations asks for an ordered pair that makes both equations true at the same time. Graphically, that pair is the intersection of the two lines. Algebraically, substitution and elimination are two reliable paths to the same point. Substitution is usually efficient when one equation already isolates a variable. Elimination is often cleaner when coefficients can be matched quickly so one variable cancels after adding or subtracting equations.
The structure of the lines tells you how many solutions exist. Different slopes usually mean one intersection, so one solution. Same slope with different intercepts gives parallel lines and no solution. If equations describe the same line, every point on that line works, so there are infinitely many solutions. In application problems, do not stop at a number pair. Interpret what each coordinate means, keep units attached, and check whether negative values or non-integers make sense in the situation before finalizing your answer.
Key terms
- system of equations
- A set of equations considered together, seeking shared solutions.
- intersection
- The point where two graphs meet, representing a common solution.
- substitution
- A method that replaces one variable using an equivalent expression from another equation.
- elimination
- A method that adds or subtracts equations to remove one variable.
Solve by elimination
Solve the system 3x + 2y = 16 and x - 2y = 0.
- Add the equations so y terms cancel: (3x + 2y) + (x - 2y) = 16 + 0.
- Simplify to 4x = 16, so x = 4.
- Substitute x = 4 into x - 2y = 0 to get 4 - 2y = 0.
- Solve -2y = -4, so y = 2.
- Check in the first equation: 3(4) + 2(2) = 12 + 4 = 16, correct.
A common misconception
Claim: If each equation is true by itself, any x and y from either equation can be paired as a system solution.
Correction: A system solution must satisfy both equations simultaneously, so values must be the same ordered pair in each equation.
Lessons in this chapter
- Graphing systemsUse intersection points as shared solutions.
- SubstitutionReplace one variable with an equivalent expression.
- EliminationCombine equations to remove one variable.
- Systems in contextInterpret a solution with units and real constraints. Read the full guide →
Study task
Chapter checkpoint
Solve x + y = 9 and x - y = 3.
Add the equations to get 2x = 12, so x = 6. Substitute to get y = 3.