Ch 7.1 – Solving Linear Systems by Graphing Algebra 1 Ch 7.1 – Solving Linear Systems by Graphing

Before we begin… In previous chapters you learned how to transform and graph linear equations using slope-intercept… In this chapter, we will use that information to solve systems of linear equations in two variables… The solution to a system of linear equations is an ordered pair or pairs that make both statements true… To be successful here it is important to lay out your work in a logical sequential manner and not skip any steps.

Process The steps below are the process to solving a system of linear equations by graphing Write each equation in a format that is easy to graph. Graph both equations on the same coordinate plane Determine the coordinates of the point of intersection Check the coordinates algebraically by substituting them into each equation of the original system of linear equations. Note: In order for the ordered pair to be a solution, after substituting, BOTH equations MUST be a true statement!

Example #1 Solve the linear system graphically. Check your solution algebraically. x + y = -2 2x – 3y = -9 The first step is to write the equations in a format that is easy to graph…

Step 1 – Rewrite Equations x + y = -2 -x -x y = -x - 2 2x – 3y = -9 -2x -2x - 3y = -2x – 9 -3 -3 y = 2/3x + 2 m = -1 b = -2 m = 2/3 b = +2 Now that you have both equations in slope-intercept form, the next step is to graph the equations on the same coordinate plane and determine the point where the 2 lines intersect

Step 2 – Graph Equations y = -x - 2 y = 2/3x + 2 y x m = -1, b = -2

Step 3 – Determine Intersection y x y = 2/3x + 2 After graphing, it appears that the 2 lines intersect at the point (-3, 1) y = -x - 2 The next step is to substitute this point into both equations and solve algebraically

Step 4 – Check Solution Algebraically Substitute the Ordered pair (-3, 1) into each equation and solve algebraically. x + y = - 2 -3 + 1 = - 2 -2 = - 2 √ 2x – 3y = -9 2(-3) – 3(1) = -9 -6 – 3 = - 9 -9 = - 9 √ TRUE TRUE Because (-3, 1) is the solution to BOTH equations it is the solution of this system of linear equations

Comments The method used to solve the example is called the graph & check method. Sometimes you are given a graph with the system of linear equations already graphed. If that is the case, determine the intersection point and substitute the ordered pair into both equations to determine if it is the solution

Comments In this example you could probably figure out the solution in your head…However, that is not the most efficient way to work with these types of problems… Additionally, the graph and check method is not the only way to solve a system of linear equations… We will explore more methods as we go through this chapter…

Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn Determine if the ordered pair is a solution to the system of linear equations 3x – 2y = 11 -x + 6y = 7 (5, 2) 6x – 3y = -15 2x + y = -3 (-2,1) x + 3y = 15 4x + y = 6 (3, -6) -5x + y = 19 x – 7y = 3 (-4, -1) -15x + 7y = 1 3x – y = 1 (3 , 5)

Your Turn Graph & check to solve the linear system y = -x + 3 5x + 4y = 16 y = -16 3x + 6y = 15 -2x + 3y = -3 1/5x + 3/5y = 12/5 -1/5x + 3/5y = 6/5

Your Turn Solutions Solution Not a solution (1, 2) (2,0) (16,-16) (3,1) (3,3)

Summary A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about solving linear systems by graphing. Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words Please be advised – I will not give any credit for work submitted: Without a complete heading Without showing work for the your turn problems Without a summary in your own words…