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.
What is a System of Linear Equations?A system of linear equations is simply two or more linear equationsusing the same variables.We will only be dealing with systems of two equations using twovariables, x and y.If the system of linear equations is going to have a solution, thenthe solution will be an ordered pair (x , y) where x and y makeboth equations true at the same time.We will be working with the graphs of linear systems and how to findtheir solutions graphically.
How to Use Graphs to Solve Linear Systems yConsider the following system: x – y = –1 x + 2y = 5Using the graph to the right, we cansee that any of these ordered pairswill make the first equation true since xthey lie on the line. (1 , 2)We can also see that any of thesepoints will make the second equationtrue.However, there is ONE coordinatethat makes both true at the sametime…The point where they intersect makes both equations true at the same time.
How to Use Graphs to Solve Linear Systems yConsider the following system: x – y = –1 x + 2y = 5We must ALWAYS verify that yourcoordinates actually satisfy bothequations. x (1 , 2)To do this, we substitute thecoordinate (1 , 2) into bothequations. 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.
Graphing to Solve a Linear SystemSolve 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.
Graphing to Solve a Linear SystemSolve the following system by graphing: y 3x + 6y = 15 –2x + 3y = –3Using the slope intercept form of theseequations, we can graph them carefully xon graph paper. (3 , 1) y =- 1 x + 5 2 2 y = 2 x- 1 3 Label theStart 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!
Graphing to Solve a Linear System Lets 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 thatintercept form. each equation looks like y = mx + b.Step 2: Graph both equations on the Use the slope and y - intercept forsame 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 theintersect. solution!Step 4: Check to make sure your Substitute the x and y values into bothsolution makes both equations true. equations to verify the point is a solution to both equations.
Graphing to Solve a Linear System Lets 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 xStep 2: Graph both equations on thesame coordinate plane.Step 3: Estimate where the graphsintersect. LABEL the solution! 2( 1) + 2 ( 1 ) = 2 +1 = 3Step 4: Check to make sure your 2solution makes both equations true. 1- 4 ( 1 ) = 1- 2 = - 1 2
•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)