6.1 Solving Systems by Graphing:

System of Linear Equations: Two or more linear equations 6.1 Solving Systems by Graphing: 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:

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.

SOLUTION: 1) Write the equations in slope-intercept form (y=mx+b)

SOLUTION: 2) Graph the equations

SOLUTION: 3) Find the solution 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.

YOU TRY IT:What is the solution of the system? Use a graph to check your answer.

SOLUTION: 2) Graph the equations

SOLUTION: 3) Find the solution 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.

CLASSWORK:Page 330-332 Problems: 25, 26, 35, 36.

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.

SOLUTION: 2) Graph the equations 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: 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 Cost 100 90 80 70 60 50 40 30 20 Months 10 4 5 6 1 2 3

SOLUTION: 5) Interpret the solution. Cost Notice: These lines intersect at at(5, 70). 100 90 80 70 60 50 40 30 This means that the two satellite services will cost the same in 5 months and $70. 20 Months 10 4 5 6 1 2 3

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 Months 1 4 5 6 1 2 3

SOLUTION: 5) Interpret the solution. Weight Notice: These lines intersect at at(4, 10) 10 9 8 7 6 5 4 3 This means that the two Alligators will Weight 10 lbs after 4 months. 2 Months 1 4 5 6 1 2 3

CLASSWORK:Page 330-332 Problems: 19, 20, 21.

6.1 Solving Systems by Graphing: