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3.1 solving systems graphically
- 1. 3.1 SOLVING Systems of Linear Equations BY GRAPHING Today’s objectives: 1. I will check solutions of a linear system. 2. I will graph and solve systems of linear equations in two variables.
- 2. 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.
- 3. How to Use Graphs to Solve Linear Systems y 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 x they lie on the line. (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 time…The point where they intersect makes both equations true at the same time.
- 4. How to Use Graphs to Solve Linear Systems y Consider the following system: x – y = –1 x + 2y = 5 We must ALWAYS verify that your coordinates actually satisfy both equations. x (1 , 2) To do this, we substitute the coordinate (1 , 2) into both equations. x – y = –1 x + 2y = 5 (1) – (2) = –1 (1) + 2(2) = Since (1 , 2) makes both equations 1+4=5 true, then (1 , 2) is the solution to the system of linear equations.
- 5. Graphing to Solve a Linear System Solve the following system by graphing: 3x + 6y = 15 Start with 3x + 6y = 15 –2x + 3y = –3 Subtracting 3x from both sides yields 6y = –3x + 15 While there are many different Dividing everything by 6 gives us… ways to graph these equations, we will be using the slope - intercept y =- 1 2 x+ 5 2 form. Similarly, we can add 2x to both sides and then divide everything by To put the equations in slope 3 in the second equation to get intercept form, we must solve both equations for y. y = 2 x- 1 3 Now, we must graph these two equations.
- 6. 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 x on graph paper. (3 , 1) y =- 1 x + 5 2 2 y = 2 x- 1 3 Label the Start at the y - intercept, then use the slope. 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!
- 7. 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 - Solve both equations for y, so that intercept form. each equation looks like y = mx + b. Step 2: Graph both equations on the Use the slope and y - intercept for same coordinate plane. each equation in step 1. Be sure to use a ruler and graph paper! Step 3: Estimate where the graphs This is the solution! LABEL the intersect. solution! Step 4: Check to make sure your Substitute the x and y values into both solution makes both equations true. equations to verify the point is a solution to both equations.
- 8. Graphing to Solve a Linear System Let's do ONE more…Solve the following system of equations by graphing. 2x + 2y = 3 y x – 4y = -1 LABEL the solution! Step 1: Put both equations in slope - intercept form. (1, 1) 2 y =- x + 3 2 y = 1 x+ 1 4 4 x Step 2: Graph both equations on the same coordinate plane. Step 3: Estimate where the graphs intersect. LABEL the solution! 2( 1) + 2 ( 1 ) = 2 +1 = 3 Step 4: Check to make sure your 2 solution makes both equations true. 1- 4 ( 1 ) = 1- 2 = - 1 2
- 9. •If the lines cross once, there will be one solution. (Consistent & Independent) •If the lines are parallel, there will be •no solution. (Inconsistent) •If the lines are the same, there will be infinitely many solutions. (Consistent & Dependent)