Lesson 7.1 Solving Systems of Equations by Graphing

What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair (x, y). When graphing, you will encounter three possibilities.

1) Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) (1,2)

What is the solution of the system graphed below? (2, -2) (-2, 2) No solution Infinitely many solutions

2) Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts.

3) Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

What is the solution of this system? y=3x – 8 y = 3x -8 (3, 1) (4, 4) No solution Infinitely many solutions

Lets try one: Find the solution to the following system: y = -2x+4 y = x -2 m=-2 b=4 m=1 b=-2 Step 1:Graph both equations Step 2: Find Where the lines intersect? The point is (2, 0)

Check your answer! Y=-2x+4 2(2) + 4 = 0 y = x-2 (2) – 2 = 0 To check your answer, substitute the point back into both equations. Y=-2x+4 2(2) + 4 = 0 y = x-2 (2) – 2 = 0 Nice job…let’s try another!

You try: Find the solution to the following system y = 2x – 1 y = –x + 5 Graph both equations Where do they intersect? The point is (2, 3)

Graph the equations. y = 2x – 3 m = 2 and b = -3 y = 2x + 1 Where do the lines intersect? No solution! Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t have to graph them!

Solving a system of equations by graphing. Let's summarize! There are 3 steps to solving a system using a graph. Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper! Step 1: Graph both equations. This is the solution! LABEL the solution! Step 2: Do the graphs intersect? Substitute the x and y values into both equations to verify the point is a solution to both equations. Step 3: Check your solution.