Systems of Equations: Graphing Method

Find solutions to linear systems by graphing. Math 10-C Alberta mathematics curriculum.

Lesson 5.1 of Systems of Equations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• What is a System of Equations?. Learn what a system of linear equations is and what it means to be a solution. Practice verifying ordered pairs by substituting into both equations, and see why a point must satisfy every equation in the system to count as the solution.
• Solving by Graphing. Graph two lines on the same coordinate plane and read off the point of intersection. Work through converting equations to slope-intercept form, reading a system from a pre-drawn graph, and using algebra to find an exact intersection when the graph is not easy to read.
• Properties of Systems — How Many Solutions?. Discover the three possible outcomes for a system: one solution, no solution, or infinitely many solutions. Learn to predict the outcome by comparing slopes and y-intercepts without graphing, and practice writing systems that have a specified number of solutions.

Inside the lesson — a free preview

Definition

System of Linear Equations. A system of linear equations consists of two or more linear equations with the same variables.

For example:

x - 2y = 12 and 3x - 2y = 4

The solution to a system is an ordered pair (x, y) that satisfies both equations at the same time. On a graph, it is the point of intersection of the two lines.

Verifying a Solution

How to Verify. To check whether a point is a solution, substitute it into both equations. It must work in both — if it fails either one, it is not a solution.

A point that satisfies one equation but not the other is not a solution to the system.

Example 1 — Is (-4, -8) a solution to x - 2y = 12 and 3x - 2y = 4?. Equation 1: x - 2y = 12

(-4) - 2(-8) = -4 + 16 = 12 ✓

Equation 2: 3x - 2y = 4

3(-4) - 2(-8) = -12 + 16 = 4 ✓

Both equations are satisfied, so (-4, -8) is a solution.

Example 2 — Is (2, 5) a solution to 3x - y = 2 and x + 4y = 32?. Equation 1: 3x - y = 2

3(2) - 5 = 6 - 5 = 1 ≠ 2 ✗

It fails the first equation, so (2, 5) is not a solution. There is no need to check the second equation — failing one is enough.

Example 3 — Is (0, -4) a solution to 2x + 3y = -12 and 4x - 3y = -6?. Equation 1: 2x + 3y = -12

2(0) + 3(-4) = 0 - 12 = -12 ✓

Equation 2: 4x - 3y = -6

4(0) - 3(-4) = 0 + 12 = 12 ≠ -6 ✗

(0, -4) is not a solution — it satisfies the first equation but not the second.

Worked example: Solve 2x + y = 2 and x - y = 7

Example 4 — Convert and Graph. Step 1 — Convert to slope-intercept form.

Equation 1: 2x + y = 2 → y = -2x + 2

Equation 2: x - y = 7 → y = x - 7

Step 2 — Graph both lines and find the intersection.

The two lines intersect at (3, -4).

Verify: (3, -4). Equation 1: 2x + y = 2

2(3) + (-4) = 6 - 4 = 2 ✓

Equation 2: x - y = 7

3 - (-4) = 3 + 4 = 7 ✓

Solution: (3, -4)

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