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6.1 Solving Systems by Graphing: System of Linear Equations: Two or more linear equations Solution of a System of Linear Equations: Any ordered pair that makes all the equations in a system true. Trend line: Line on a scatter plot, drawn near the points, that shows a correlation Consistent: System of equations that has at least one solution. 1) Could have the same or different slope but they intersect. 2) The point where they meet is a solution Consistent Independent: System of equations that has EXACTLY one solution. 1) Have different slopes 2) Only intersect once 3) The point of intersection is the solution. Consistent Dependent: System of equations that has infinitely many solutions. 1) Have same slopes 2) Same y-intercepts 3) Each point is a solution. Inconsistent: System of equations that has no solutions. 1) Have same slopes 2) different y-intercepts 3) No solutions Remember: Remember: GOAL: SOLVING A SYSTEM BY GRAPHING: To solve a system by graphing we must: 1) Write the equations in slope-intercept form (y=mx+b) 2) Graph the equations 3) Find the point of intersection 4) Check Ex: What is the solution of the system? Use a graph to check your answer. 2 x y 4 y x2 SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b) 2 x y y 4 x2 y 2x 4 y x2 SOLUTION: 2) Graph the equations y x2 y 2x 4 SOLUTION: 3) Find the solution y 2x 4 y x 2 Looking at the graph, we see that these two equations intersect at the point : (-2, 0) SOLUTION: 4) Check We know that (-2,0) is the solution from our graph. y 2x 4 0 2( 2 ) 4 y x2 0 0 4 4 0 2 2 0 0 0 TRUE TRUE YOU TRY IT: What is the solution of the system? Use a graph to check your answer. x y 2 y 2 3x SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b) x y 2 y2 3x y x2 y 3x 2 SOLUTION: 2) Graph the equations y 3x 2 y x2 SOLUTION: 3) Find the solution y x 2 y 3x 2 Looking at the graph, we see that these two equations intersect at the point : (2,4) SOLUTION: 4) Check We know that (2,4) is the solution from our graph. y x2 4 2 2 y 3x 2 4 4 4 TRUE 3( 2 ) 2 4 62 4 4 TRUE SYSTEM WITH INFINITELY MANY SOLUTIONS: Using the same procedure we can see that sometimes the system will give us infinitely many solutions (any point will make the equations true). Ex: What is the solution to the system? Use a graph. 2 y x 2 1 y x1 2 SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b) 2 y x 2 2y x 2 y 1 y 2 3x y x 1 2 1 2 x1 SOLUTION: 2) Graph the equations 1 y x1 2 1 y x1 2 Notice: Every point of one line is on the other. SYSTEM WITH NO SOLUTIONS: Using the same procedure we can see that sometimes the system will give us infinitely many solutions (any point will make the equations true). Ex: What is the solution to the system? Use a graph. − + = − = SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b) − + = → = + − = → − = − + = − SOLUTION: 2) Graph the equations = + = − Notice: These lines will never intersect. NO SOLUTIONS. WRITING A SYSTEM OF EQUATIONS: Putting ourselves in the real world, we must be able to solve problems using systems of equations. Ex: One satellite radio service charges $10.00 per month plus an activation fee of $20.00. A second service charges $11 per month plus an activation fee of $15. For what number of months is the cost of either service the same? SOLUTION: Looking at the data we must be able to do 5 things: 1) Relate- Put the problem in simple terms. Cost = service charge + monthly dues 2) Define- Use variables to represent change: Let C = total Cost Let x = time in months SOLUTION: (continue) 3) Write- Create two equations to represent the events. Satellite 1: C = $10 x + $20 Satellite 2: C = $11 x + $15 4) Graph the equations: Remember to put them in slope/intercept form (y = mx + b) The two equations are already in y=mx+b form. SOLUTION: 4) Continue = + = + 100 Cost 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 Months SOLUTION: 5) Interpret the solution. Cost Notice: 100 90 These lines 80 70 intersect at 60 = + 50 at (5, 70). 40 = + 30 This means 20 10 that the two 1 2 3 4 5 6 satellite services Months will cost the same in 5 months and $70. YOU TRY IT: Scientists studied the weights of two alligators over a period of 12 months. The initial weight and growth rate of each alligator are shown below. After how many months did the two alligators weight the same? SOLUTION: Looking at the data, Here are the 5 things we must do: 1) Relate- Put the problem in simple terms. Total Weight = initial weight + growth per month. 2) Define- Use variables to represent change: Let W = Total weight Let x = time in months SOLUTION: (continue) 3) Write- Create two equations to represent the events. Alligator 1: W = 1.5x + 4 Alligator 2: W= 1.0 x + 6 4) Graph the equations: Remember to put them in slope/intercept form (y = mx + b) The two equations are already in y=mx+b form. SOLUTION: 4) Continue W = . + = + Weight 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 Months SOLUTION: 5) Interpret the solution. Weight 10 Notice: 9 8 These lines 7 intersect at 6 5 at (4, 10) 4 3 This means 2 1 that the two 1 2 3 4 5 6 Alligators will Months Weight 10 lbs after 4 months. VIDEOS: Solve by Graphing https://www.khanacademy.org/math/algebra/syst ems-of-eq-and-ineq/fast-systems-ofequations/v/solving-linear-systems-by-graphing CLASSWORK: Page 363-365 Problems: As many as needed to master the concept.