Systems of Equations

Grade 9 · Math · Study Guide

Learn to solve systems of two linear equations using substitution, elimination, and graphing methods.

A system of equations is a set of two or more equations with the same variables. A solution is an ordered pair (x, y) that satisfies every equation in the system simultaneously. Graphically, the solution is the point where the lines intersect.

The substitution method works by solving one equation for one variable, then substituting that expression into the other equation. For example, given y = 2x + 1 and 3x + y = 16, substitute: 3x + (2x + 1) = 16 → 5x + 1 = 16 → x = 3, then y = 7. The solution is (3, 7).

The elimination method adds or subtracts the equations to eliminate one variable. Multiply equations by constants if necessary so coefficients match. For 2x + 3y = 12 and 2x − y = 4, subtract to get 4y = 8, so y = 2; substitute back to get x = 3. Special cases: parallel lines give no solution; identical lines give infinitely many solutions.

Comparing Solution Methods
Method	Best Used When	Key Step
Graphing	Estimating or visualizing the solution	Find intersection point
Substitution	One equation is already solved for a variable	Plug expression into other equation
Elimination	Coefficients can be made equal easily	Add/subtract to cancel a variable

Key terms

System of equations: Two or more equations with the same set of variables
Solution of a system: An ordered pair (x, y) that satisfies all equations simultaneously
Substitution method: Solving one equation for a variable, then substituting into the other
Elimination method: Adding/subtracting equations to cancel one variable
Consistent system: A system with at least one solution
Inconsistent system: A system with no solution (parallel lines)

Flashcards

System of equations: Two or more equations sharing the same variables
Substitution method: Solve one equation for a variable; plug into the other equation
Elimination method: Multiply and add/subtract equations to remove one variable
Ordered pair solution: A pair (x, y) that satisfies every equation in the system
Inconsistent system: No solution exists — the lines are parallel
Dependent system: Infinitely many solutions — both equations represent the same line

Practice questions

Solve the system: y = x + 3 and y = 2x + 1
1. (2, 5)
2. (3, 6)
3. (1, 4)
4. (4, 7)
Show answer
Answer: (2, 5)
Set equal: x + 3 = 2x + 1 → x = 2; then y = 5.
A system of two parallel lines has infinitely many solutions.
Show answer
Answer: False
Parallel lines never intersect, so there is no solution.
When solving by elimination, you add or subtract equations to ___ one variable.
Show answer
Answer: eliminate
The goal of elimination is to add/subtract equations so one variable cancels out.
Solve: x + y = 10 and x − y = 4
1. (7, 3)
2. (6, 4)
3. (8, 2)
4. (5, 5)
Show answer
Answer: (7, 3)
Add equations: 2x = 14 → x = 7; substitute: 7 + y = 10 → y = 3.
Which ordered pair is a solution to both 2x + y = 7 and x − y = 2?
1. (2, 3)
2. (3, 1)
3. (4, −1)
4. (1, 5)
Show answer
Answer: (3, 1)
Add equations: 3x = 9 → x = 3; substitute: 3 − y = 2 → y = 1.

Fun facts

Systems of equations are used in economics to find market equilibrium — the price where supply equals demand.
A 2×2 system of linear equations can have exactly 0, 1, or infinitely many solutions — never exactly 2.

Get a personalized course on Systems of Equations built around your child.