1. Advanced Algebra Notes
Section 3.1
Solving Linear Systems by Graphing

2. Solving Linear Systems by Graphing
System of two linear equations
A in two variables x and y, also called linear system, consists of two equations that can be written in the following form. When we talk about finding a we are looking for the ordered pair (x,y) that will satisfy each equation. It must work with both!
Ax + By = C
Dx + Ey = F
solution

3. Example 1
Graph the linear system and estimate the solution. Then check the solution algebraically. 4x+y=8 2x-3y=18
𝑦=−4𝑥+8
−3𝑦=−2𝑥+18
𝑦= −2𝑥+18 −3
(3,−4)
𝑦= 2 3 𝑥−6

4. Example 2 INFINITE SOLUTIONS
Graph the linear system and estimate the solution. 4x-3y=8 8x-6y=16
−3𝑦=−4𝑥+8
𝑦= 4 3 𝑥− 8 3
INFINITE SOLUTIONS
−6𝑦=−8𝑥+16
−3𝑦=−4𝑥+8
𝑦= 4 3 𝑥− 8 3

5. Example 3 𝑦=−2𝑥+4 PARALLEL 𝑦=−2𝑥+1 NO SOLUTION
Graph the linear system and estimate the solution 2x+y=4 2x+y=1
𝑦=−2𝑥+4
PARALLEL
𝑦=−2𝑥+1
NO SOLUTION
When lines have the same slope
they are parallel. They will not cross!
They will not have a solution!

6. Example 4
Graph the linear system and estimate the solution 5x-2y=-10 2x-4y=12
−2𝑦=−5𝑥−10
𝑦= 5 2 𝑥+ 10 2
𝑦= 5 2 𝑥+5
−4𝑦=−2𝑥+12
𝑦= 1 2 𝑥−3
(−4,−5)

7. Closing: How do you solve a system of linear equations graphically?
Our solution is the point where our lines
intersect. It is a coordinate point (𝑥,𝑦)
If our lines do not cross, they are parallel
meaning they have no solution.
If our lines are the same line, they are
Intersect at every point meaning they
have infinite solutions.
Our solution has to work for both equations!!!