3.1 Solving Linear Systems by Graphing

10/1/12

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.

VocabularySystem of 2

Linear Equations:

A system consisting of two linear equations in two variables.Ex: 6x – 2y = 8 3x – y = 4

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

-2x + y = -2 4x – 2y = 3Substitute 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

Solve the system by graphing. Then check your solution.

3y – x= +

9y 2x= +

Solve a System by GraphingExample 1

ANSWER ( )2, 5–

y = - x + 3 5= -(-2) + 3 5= 5

y = 2 x + 9 5 = 2(-2) + 95 = -4 + 95 = 5

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

Example 2

ANSWER ( )2, 3

Solve a System by Graphing

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

33x – y =

8x + 2y =

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

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

Example 2 Solve a System by Graphing

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

Equation 1 Equation 2

33x – y = 8x + 2y =

( )2 33 – 3 =? 2 8+ =?( )32

36 – 3 =? 2 8+ =

?6

33 = 88 =

ANSWER ( ).2, 3The solution of the system is

Extra Example

1. x 3y =– 1

x y =+ 1– –

ANSWER ( )1, 0

Solve the system by graphing. Then check your solution.

Checkpoint

2. =

2x 3y =– 6

ANSWER ( )6, 2

Solve a System by Graphing

Solve the system by graphing. Then check your solution.

2+ 4yx–

Homework WS 3.1

Number of Solutions

1 solution: the lines have different slopes

Infinitely many solutions:the lines have the same equation.

No solution:the lines are parallel (same slope)

Systems with Many or No SolutionsExample 3

Tell how many solutions the linear system has.

a. 1=y–2x

+ =2y–4x 2–

b.

+ =2yx 1

+ =2yx 4

12

2 2 2

242

4 4

224

xy

xy

x x

yx

12

1- 1- 1-

12

2 2

12

xy

xy

x -x

yx

22

1

2 2 2

42

42

xy

xy

-x x

yx

2

1

2

1

2 2 2

12

12

xy

xy

-x x

yx

Infinitely many solutions:the lines have the same equation.

No solution:the lines are parallel (same slope)

Tell how many solutions the linear system has.

Checkpoint

ANSWER 0

Write and Use Linear Systems

4. 5=4y–x

+ =4y–x 5–

3. + =3y2x 1

+ =6y4x 3

ANSWER infinitely many solutions

ANSWER 1

5. 5=5y–x

+ =5yx 5