The document discusses using graphs to solve systems of linear equations. It explains that a system of linear equations involves two or more linear equations using the same variables. The solution to a system is the point where the graphs of the equations intersect, making both equations true. It then walks through an example of putting two equations into slope-intercept form and graphing them to find the solution point, which must be verified by substituting into the original equations. Finally, it summarizes the four steps to graphically solving a linear system.

1. Systems of Linear Equations Using a Graph to Solve

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3. What is a System of Linear Equations? A system of linear equations is simply two or more linear equations using the same variables. We will only be dealing with systems of two equations using two variables, x and y . If the system of linear equations is going to have a solution, then the solution will be an ordered pair ( x , y ) where x and y make both equations true at the same time. We will be working with the graphs of linear systems and how to find their solutions graphically.

4. How to Use Graphs to Solve Linear Systems Consider the following system: x – y = –1 x + 2 y = 5 Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line. We can also see that any of these points will make the second equation true. However, there is ONE coordinate that makes both true at the same time… The point where they intersect makes both equations true at the same time. x y (1 , 2)

5. x – y = –1 x + 2 y = 5 How to Use Graphs to Solve Linear Systems Consider the following system: We must ALWAYS verify that your coordinates actually satisfy both equations. To do this, we substitute the coordinate (1 , 2) into both equations. x – y = –1 (1) – (2) = –1  Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations. x + 2 y = 5 (1) + 2(2) = 1 + 4 = 5  x y (1 , 2)

6. Graphing to Solve a Linear System While there are many different ways to graph these equations, we will be using the slope – intercept form . To put the equations in slope intercept form, we must solve both equations for y . Start with 3 x + 6 y = 15 Subtracting 3 x from both sides yields 6 y = –3 x + 15 Dividing everything by 6 gives us… Similarly, we can add 2 x to both sides and then divide everything by 3 in the second equation to get Now, we must graph these two equations Solve the following system by graphing: 3 x + 6 y = 15 – 2 x + 3 y = –3

7. Graphing to Solve a Linear System Using the slope intercept forms of these equations, we can graph them carefully on graph paper. Start at the y – intercept, then use the slope. Lastly, we need to verify our solution is correct, by substituting (3 , 1). Solve the following system by graphing: 3 x + 6 y = 15 – 2 x + 3 y = –3 x y Label the solution! (3 , 1) Since and , then our solution is correct!

8. Graphing to Solve a Linear System Let's summarize! There are 4 steps to solving a linear system using a graph. Step 1 : Put both equations in slope – intercept form Step 2 : Graph both equations on the same coordinate plane Step 3 : Estimate where the graphs intersect. Step 4 : Check to make sure your solution makes both equations true. Solve both equations for y , so that each equation looks like y = mx + b . Use the slope and y – intercept for each equation in step 1. Be sure to use a ruler and graph paper! This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations.

9. Graphing to Solve a Linear System Step 1 : Put both equations in slope – intercept form Step 2 : Graph both equations on the same coordinate plane Step 3 : Estimate where the graphs intersect. LABEL the solution! Step 4 : Check to make sure your solution makes both equations true. Let's do ONE more…Solve the following system of equations by graphing. 2 x + 2 y = 3 x – 4 y = –1 x y LABEL the solution!