Advanced Algebra Notes Section 3.1 Solving Linear Systems by Graphing

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

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

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

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!

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)

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!!!