9th grade math - Algebra

Solving Systems of Linear Equations

How to Work with Two Lines at Once

If you’ve studied our previous articles on Forms of Linear Equations, Slope and Rate of Change, and Graphing Linear Equations, you’re ready for this next important step: solving systems of linear equations.

A system of linear equations is a set of two or more linear equations with the same variables. In most cases, you’ll be given two equations and asked to find the values of x and y that make both equations true at the same time.

What Is a Solution?

A solution to a system is a point (x, y) that satisfies both equations. Graphically, this point is where the two lines intersect.

Example:

Equation 1: y = 2x + 1
Equation 2: y = -x + 4

If you graph these, they intersect at (1, 3). That means:

  • y = 2(1) + 1 = 3 ✔️
  • y = -1 + 4 = 3 ✔️

So, (1, 3) is the solution!

Methods for Solving Systems

1. Graphing

This method involves plotting both lines on the same coordinate plane and identifying the intersection point.

Steps:

  1. Write both equations in slope–intercept form: y = mx + b
  2. Graph each line using its slope and y-intercept
  3. Find the intersection point

Pros: Great for visual learners

Cons: Less accurate with fractions or decimals

2. Substitution

Use when one equation is already solved for a variable. Substitute that expression into the other equation. 

Equation 1: y = 3x + 2
Equation 2: 2x + y = 16

Step 1: Substitute y = 3x + 2 into Equation 2:

2x + (3x + 2) = 16
5x + 2 = 16
5x = 14
x = 14/5

Step 2: Plug x into Equation 1:

y = 3(14/5) + 2 = 52/5

Solution: (14/5, 52/5)

Pros: Exact answers

Cons: Requires careful algebra

3. Elimination

Also called the addition method, this technique eliminates one variable by adding or subtracting the equations. 

Equation 1: 3x + 2y = 16
Equation 2: 2x - 2y = 4

Step 1: Add both equations:

(3x + 2y) + (2x - 2y) = 16 + 4
5x = 20
x = 4

Step 2: Plug x = 4 into Equation 1:

3(4) + 2y = 16
12 + 2y = 16
2y = 4
y = 2

Solution: (4, 2)

Pros: Fast with well-aligned equations

Cons: Needs clean organization

Types of Solutions

Not every system has just one answer. Here are the three possible outcomes:

  • One Solution: Lines intersect at one point (Independent system)
  • No Solution: Lines are parallel and never meet (Inconsistent system)
  • Infinite Solutions: Both equations describe the same line (Dependent system)

Examples:

  • No Solution: y = 2x + 3 and y = 2x – 5 (same slope, different y-intercepts)
  • Infinite Solutions: y = -x + 2 and -2y = 2x – 4 (same line, written differently)

Real-World Connections

Systems of equations help solve practical problems, such as:

  • Comparing phone or subscription plans
  • Finding when two moving objects meet
  • Solving business profit and cost models

Building on What You Know

This topic ties everything together:

As you continue learning, you’ll explore systems of linear inequalities, which show regions instead of just points. But first, make sure you’re confident solving systems with lines. Try practicing all three methods — and check your solutions by plugging them back into both equations.

You’ve come a long way in algebra. Keep building those skills one equation at a time!