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 42

x yy x

SOLUTION:

2 2 y y xx

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

22 44 yy xx

SOLUTION: 2 y x

2) Graph the equations 2 4 y x

SOLUTION: 2 y x

3) Find the solution 2 4 y x

Looking at the graph, we see that these two equations intersect at the point : (-2, 0)

SOLUTION:

2 y x

4) Check

2 4 y x

We know that (-2,0) is the solution from our graph.

2( ) 40 2

0 4 4 0 0 TRUE

0 2 2 0 0 TRUE

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

22 3

x yy x

SOLUTION:

32 3 2 yy xx

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

2 2 y y xx

SOLUTION: 3 2 y x

2) Graph the equations 2 y x

SOLUTION: 3 2 y x

3) Find the solution 2 y x

Looking at the graph, we see that these two equations intersect at the point : (2,4)

SOLUTION:

3 2 y x

4) Check

2 y x

We know that (2,4) is the solution from our graph.

4 22

4 4 TRUE

4 3( 22) 4 6 2

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 2

1 12

y x

y x

SOLUTION:

1 122

3

y xy x

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

12 2 12 22

y x y xy x

SOLUTION:

1 12

y x

2) Graph the equations

1 12

y x

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. −𝟐 𝒙+𝒚=𝟐

𝟐 𝒙−𝒚=𝟏

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

−𝟐 𝒙+𝒚=𝟐→ 𝐲=𝟐𝐱+𝟐

𝟐 𝒙−𝒚=𝟏→− 𝒚=−𝟐 𝒙+𝟏

SOLUTION: 2) Graph the equations

Notice: These lines will never intersect. NO SOLUTIONS.

𝒚=𝟐 𝒙+𝟐𝒚=𝟐 𝒙−𝟏

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

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.

Satellite 2: C = $11 x + $15

SOLUTION: 4) Continue𝒚=𝟏𝟎𝒙+𝟐𝟎 𝒚=𝟏𝟏𝒙+𝟏𝟓

Cost

1 2 3 4

102030405060708090100

5 6Months

SOLUTION: 5) Interpret the solution.

Notice: These lines intersect at at (5, 70). Co

st

1 2 3 4

102030405060708090100

5 6

This means that the two satellite services will cost the same in 5 months and $70.

Months

𝒚=𝟏𝟏𝒙+𝟏𝟓𝒚=𝟏𝟎𝒙+𝟐𝟎

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

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.

Alligator 2: W= 1.0 x + 6

SOLUTION: 4) ContinueW 𝒚=𝟏 𝒙+𝟔

Wei

ght

1 2 3 4

12345678910

5 6Months

SOLUTION: 5) Interpret the solution.

Notice: These lines intersect at at (4, 10) This means that the two Alligators will Weight 10 lbs after 4 months.

Months

Wei

ght

1 2 3 4

12345678910

5 6

VIDEOS: Solve by Graphing

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/v/solving-linear-systems-by-graphing

CLASSWORK:

Page 363-365

Problems: As many as needed to master the

concept.