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Solving Linear Systems by Graphing

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Solving Linear Systems by Graphing

AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

LA, 6.0: Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions

Objectives

Key Words

Solve a system of linear equations in two variables by graphing

System of two linear equations

Solution of a system of two linear equations

Identify the slope of an equation

Solve linear equations

Tell whether the point is a solution of the equation

Graph the equations

- Take 5 minutes to discuss with someone next to you.
- Things you may want to ask:
- When have you compared the price of two items?
- Have you ever had to make a choice between to objects or situations?
- How much money did you spend the last time you were on the bus?

Can you think of when you will use it?

We are going to visit the following site for practice:

http://www.classzone.com/cz/books/algebra_2_cs/resources/applications/animations/html/alg207_ch03_pg155.html

- Take a few moments to think about how many solutions we can possibly have given two random equations.
- Things you may want to ask your partner:
- Do we need to have a solution?
- Is there always going to be a solution?
- What if there is more than one solution?
- When will we have no solutions?

The graph of the system is a pair of lines that intersects in one point

The lines have different slopes.

The system has exactly one solution.

The graph of the system is a pair of identical lines

The lines have the same slope and the same y-intercept

The system has infinitely many solutions

The graph of the system is a pair of parallel lines, which do not intersect

The lines have the same slope and different y-intercept

The system has no solutions

Steps:

- For each equation plot two points and draw the line:
- Pick two inputs to get two outputs
- For example, pick x=0 and x=1

- Find the intersections
- You may approximate the answer

- Verify your answer is correct.
- Plug in you x and y value and check if it is true

Example 1

=

3x

–

y

3

=

x

+

2y

8

SOLUTION

Graph both equations, as shown. From the graph, you can see the lines appear to intersect at

2,3

(

).

Solve a System by Graphing

Solve the system by graphing. Then check your solution algebraically.

Equation 1

Equation 2

Example 1

?

?

=

=

Equation 1

Equation 2

(

2, 3

).

=

x

+

8

2y

?

2

8

2

+

3

–

3

3

=

The solution of the system is

?

6

–

3

3

6

=

2

8

+

=

3x

–

y

3

=

=

3

8

3

8

ANSWER

(

(

)

)

2

3

Solve a System by Graphing

You can check the solution by substituting 2 for x and 3 for y into the original equations.

Checkpoint

+

=

y

2

x

9

+

=

y

–

x

3

ANSWER

–

(

2, 5

)

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

1.

Checkpoint

–

–

+

–

=

x

y

1

=

x

3y

1

ANSWER

(

1, 0

)

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

2.

Checkpoint

–

x

+

4y

2

–

=

2x

3y

6

ANSWER

(

6, 2

)

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

3.

=

b.

–

2x

y

1

=

x

+

2y

4

=

–

–

4x

+

2y

2

=

SOLUTION

x

+

2y

1

=

a.

Because the graph of each equation is the same, each point on the line is a solution. So, the system has infinitely many solutions.

Example 2

Systems with Many or No Solutions

Tell how many solutions the linear system has.

a.

Example 2

Systems with Many or No Solutions

b.

Because the graphs of the equations are two parallel lines, the two lines have no point of intersection. So, the system has no solution.

Days in

San Diego

Days in

Anaheim

Total

Budget

SOLUTION

VERBAL

MODEL

Days in

Anaheim

Days in

San Diego

Total

vacation time

You can use a verbal model to write a system of linear equations.

+

=

Daily

cost in

San Diego

Daily

cost in

Anaheim

+

•

+

•

=

Example 3

Write and Use a Linear System

Vacation You are planning a 7-day trip to California. You estimate that it will cost $300 per day in San Diego and $400 per day in Anaheim. Your total budget for the trip is $2400. How many days should you spend in each city?

Equation 2 (total budget)

+

x

y

7

=

+

300x

400y

2400

=

Equation 1 (total vacation time)

ALGEBRAIC

MODEL

Total budget 2400

=

Example 3

Write and Use a Linear System

(days)

Days in San Diego x

LABELS

=

(days)

Days in Anaheimy

=

(days)

Total vacation time 7

=

(dollars per day)

Daily cost in San Diego 300

=

(dollars per day)

Daily cost in Anaheim 400

=

(dollars)

The lines appear to intersect at .

(

)

4,3

Example 3

Write and Use a Linear System

Graph both equations only in the first quadrant because the only values that make sense in this situation are

positive values of x and y.

Equation 1

+

+

x

y

4

3

7

=

=

Equation 2

(

)

4, 3

(

)

+

+

3

300x

400y

300

400

2400

=

=

ANSWER

The solution is . You should plan to spend 4 days in San Diego and 3 days in Anaheim.

(

)

4

Example 3

Write and Use a Linear System

CHECK

Substitute 4 for x and 3 for y in the original equations.

Checkpoint

6.

4.

2x

+

3y

1

=

0

ANSWER

4x

+

6y

3

=

1

ANSWER

5.

–

x

4y

5

=

–

–

x

+

4y

5

=

infinitely many solutions

ANSWER

–

x

5y

5

=

x

+

5y

5

=

Write and Use Linear Systems

Tell how many solutions the linear system has.

Checkpoint

7.

Vacation Your family is planning a 6-day trip to Florida. You estimate that it will cost $450 per day in Tampa and $600 per day in Orlando. Your total budget is $3000. How many days should you spend in each city?

ANSWER

4 days in Tampa and 2 days in Orlando

Write and Use Linear Systems

Tell how many solutions the linear system has.

Summary

Assignment

- How do you solve a system of linear equations graphically?
- Graph the linear equations and estimate the point where the graphs cross. Check the solution algebraically.

Pg128 #(10,16,28,36,37)

Due by the end of the class.