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3.1 Solving Linear Systems by GraphingPowerPoint Presentation

3.1 Solving Linear Systems by Graphing

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System of 2 Linear Equations:

A system consisting of two linear

equations in two variables.

Ex: 6x – 2y = 8

3x – y = 4

Solution of a system of 2 linear equations:

Is an ordered pair (x, y) that satisfies both equations.

Graphically, it’s the point where the lines intersect.

Tell whether the ordered pair (3, 4) is a solution of

-2x + y = -2

4x – 2y = 3

Substitute 3 for x and 4 for y in BOTH equations.

-2(3) + 4 = -2

- 6 + 4 = -2

4(3) – 2(4) = 3

12 – 8 = 3

Answer: Not a Solution

Tell whether the ordered pair (3, 4) is a solution of

x + 2y = 11

2x – y = 2

Substitute 3 for x and 4 for y in BOTH equations.

3 + 2(4) = 11

3 + 8 = 11

2(3) – 4 = 2

6 – 4 = 2

Answer: Solution

=

y

2

x

9

+

=

y

–

x

3

ANSWER

–

(

2, 5

)

Example 1

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

You can check the solution by substituting -2for x and 5for y into the original equations.

y= - x + 3

5= -(-2) + 3

5= 5

y = 2 x + 9

5 = 2(-2) + 9

5 = -4 + 9

5 = 5

Standard Slope Int Form

- Add or subtract the x – term on both sides of the equation.
- Divide everything by the coefficient of y if the coefficient is not 1.

Ex.

Ex.

Example 2

=

3x

–

y

3

=

x

+

2y

8

ANSWER

(

2,3

)

Solve a System by Graphing

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

In slope int. form: y = 3x - 3

In slope int. form:y = - x + 4

Example 2

?

?

=

=

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.

–

–

+

–

=

x

y

1

=

x

3y

1

ANSWER

(

1, 0

)

Solve the system by graphing. Then check your solution.

2.

ANSWER

(

2, 1

)

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

1 solution

: the lines have different slopes

No solution

:the lines are parallel (same slope)

Infinitely many solutions

:the lines have the same equation.

–

2x

y

1

=

x

+

2y

4

=

–

–

4x

+

2y

2

=

x

+

2y

1

=

Example 3

Systems with Many or No Solutions

Tell how many solutions the linear system has.

a.

Infinitely many solutions

:the lines have the same equation.

No solution

:the lines are parallel (same slope)

3.

1.

2x

+

3y

1

=

0

ANSWER

4x

+

6y

3

=

1

ANSWER

2.

–

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 without graphing.

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