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3-1: Solving Systems by Graphing

Learn how to solve systems of linear equations using graphical methods with examples and practice exercises. Check coordinates to ensure solutions are accurate.

kimberlybailey
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3-1: Solving Systems by Graphing

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  1. 3-1: Solving Systems by Graphing

  2. Example: 3x + 2y = 2 Equation 1 x + 2y = 6 Equation 2 What is a System of Linear Equations? Definition: A system of linear equations is simply two or more linear equations using the same variables.

  3. y x (1 , 2) How to Use Graphs to Solve Linear Systems Consider the following system: x – y = –1 x + 2y = 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.

  4. Three Possible Outcomesp. 154 • Two intersecting lines • Two lines on top of each other • Two parallel lines

  5. EXAMPLE 4 Writing and Using a Linear System (p. 155)

  6. x 2.5 y = y x 1 30 + = EXAMPLE 4 Step 1: Write linear equations in standard form Equation 1 Equation 2

  7. EXAMPLE 4 Step 2: Graph both equations • Two intersecting lines = one solution • (20, 50) appears to be the solution

  8. ANSWER The solution is (20, 50). Break even point in 20 rides EXAMPLE 4 Step 4: Check your solution Point of intersection: (20,50). Substitute 20 and 50 in place of x and y in both equations: y = x + 30 y =2.5x Equation 1 checks. 50= 20+ 30 50= 2.5(20) Equation 2 checks.

  9. 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. Solve both equations for y, so that each equation looks like y = mx + b. Step 2: Graph both equations on the same coordinate plane. Use the slope and y - intercept for each equation in step 1. Be sure to use a ruler and graph paper! Step 3: Estimate where the graphs intersect. This is the solution! LABEL the solution! Step 4: Check to make sure your solution makes both equations true. Substitute the x and y values into both equations to verify the point is a solution to both equations.

  10. y x (6 , -1) Practice: Checking the Solution Page 412, #11: 4x – y = 25 -3x - 2y = -16 We must ALWAYS verify that your coordinates actually satisfy both equations. To do this, we substitute the coordinate (6 , -1) into both equations. -3x - 2y = -16 -3(6) - 2(-1) = -18 + 2 = -16  4x – y = 25 4(6) – (-1) = 24 + 1 = 25  Since (6 , -1) makes both equations true, then (6 , -1) is the solution to the system of linear equations.

  11. Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (3, – 4). You can check this algebraically as follows. EXAMPLE 1 Solve a system graphically Graph the linear system and estimate the solution. Then check the solution algebraically. 4x + y = 8 Equation 1 2x – 3y = 18 Equation 2 SOLUTION

  12. 4(3) +(–4) 8 8 = 8 2(3) – 3(– 4) 18 6 + 12 18 12 – 4 8 18 = 18 ? ? ? ? = = = = EXAMPLE 1 Solve a system graphically Equation 2 Equation 1 2x– 3y= 18 4x+ y= 8 The solution is (3, – 4).

  13. 1. 3x + 2y = – 4 x + 3y = 1 Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (–2, 1). You can check this algebraically as follows. Page 153 Example 1 GUIDED PRACTICE Graph the linear system and estimate the solution. Then check the solution algebraically. 3x + 2y = – 4 Equation 1 x + 3y = 1 Equation 2 SOLUTION

  14. (–2 ) + 3(1) 1 ? = 3(–2) +2(1) –4 –2 + 3 1 –4 = –4 –6 + 2 –4 1 = 1 ? ? ? = = = Page 153, Example 1 GUIDED PRACTICE Equation 2 Equation 1 3x+ 2y= –4 x+ 3y= 1 The solution is (–2, 1).

  15. Solve the following system by graphing: 3x + 4y = -10 -7x - y = -10 Page 142, #23 Start with 3x + 4y = -10 Subtracting 3x from both sides yields 4y = –3x -10 Dividing everything by 6 gives us… While there are many different ways to graph these equations, we will be using the slope - intercept form. Similarly, we can add 7x to both sides and then divide everything by -1 in the second equation to get To put the equations in slope intercept form, we must solve both equations for y. Now, we must graph these two equations.

  16. y Solve the following system by graphing: 3x + 6y = 15 –2x + 3y = –3 x Page 142, #23, cont. Using the slope intercept form of these equations, we can graph them carefully on graph paper. Start at the y – intercept: Note that my scale is 2 on this graph. then use the slope.

  17. Page 142, #23, cont. Can you read the solution? It looks close to (2, -4) Check to make sure.

  18. y LABEL the solution! x Graphing to Solve a Linear System Let's do ONE more…Solve the following system of equations by graphing. 2x + 2y = 3 x – 4y = -1 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.

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