Systems of Linear Equations - Teachers.Henrico Webserver
Systems of Linear Equations Using a Graph to Solve All the slides in this presentation are timed. You do not need to click the mouse or press any keys on the keyboard for the presentation on each slide to continue. However, in order to make sure the presentation does not go too quickly, you will need to click the mouse or press a key on the keyboard to advance to the next slide. You will know when the slide is finished when you see a small icon in the bottom left corner of the slide. Click the mouse button to advance the slide when you see this icon. 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. If the lines are parallel, there will be no solutions. If the lines are the same, there will be an infinite number of solutions. We will be working with the graphs of linear systems and how to find their solutions graphically. 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. y (1 , 2)
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 timeThe point where they intersect makes both equations true at the same time. x If the lines cross once, there will be one solution. If the lines are parallel, there will be no solutions. If the lines are the same, there will be an infinite number of solutions. How to Use Graphs to Solve Linear Systems Consider the following system:
x y = 1 y x + 2y = 5 We must ALWAYS verify that your coordinates actually satisfy both equations. (1 , 2) x To do this, we substitute the coordinate (1 , 2) into both equations. x y = 1 (1) (2) = 1 x + 2y = 5
(1) + 2(2) = 1+4=5 Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations. Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 2x + 3y = 3 Start with 3x + 6y = 15 Subtracting 3x from both sides yields 6y = 3x + 15 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. Dividing everything by 6 gives us y =- 1 2 x + 52 Similarly, we can add 2x to both sides and then divide everything by 3 in the second equation to get y = 23 x - 1 Now, we must graph these two equations.
Graphing to Solve a Linear System Solve the following system by graphing: y 3x + 6y = 15 2x + 3y = 3 Using the slope intercept form of these equations, we can graph them carefully on graph paper. (3 , 1) y =- 12 x + 52 y = 23 x - 1 Start at the y - intercept, then use the slope. Label the solution!
Lastly, we need to verify our solution is correct, by substituting (3 , 1). Since 3( 3) + 6( 1) = 15 and - 2 ( 3) + 3( 1) =- 3 , then our solution is correct! x 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. Graphing to Solve a Linear System Let's do ONE moreSolve the following system of equations by graphing. 2x + 2y = 3 x 4y = -1 Step 1: Put both equations in slope intercept form.
y =- x + 32 y = 14 x + 14 y LABEL the solution! ( 1 , 12 ) x 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. 2( 1) + 2 ( 12 ) = 2 +1 = 3 1- 4( 12 ) = 1- 2 =- 1
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