3.1Graph each of the following problems

1.

2.

3.

4.

5.

6.

3.1A system of linear equations is 2 or more equations that intersect at the same point or have the same solution.

You can find the solution to a system of equations in several ways. The one you are going to learn today is to find a solution by graphing. The solution is the ordered pair where the 2 lines intersect.

In order to solve a system, you need to graph both equations on the same coordinate plane and then state the ordered pair where the lines intersect.

Lines intersect at one point:consistent and independent

Lines coincide;consistent and dependent

Lines are parallel;inconsistent

Classifying Systems

• Consistent – a system that has at least one solution

• Inconsistent – a system that has no solutions

• Independent – a system that has exactly one solution

• Dependent – a system that has infinitely many solutions

GUIDED PRACTICE

From the graph, the lines appear to intersect at (–2, 1).

Graph each system and then estimate the solution.

1. 3x + 2y = -4 x + 3y = 1

2. 4x – 5y = -10 2x – 7y = 4

3x + 2y = -4

2y = -3x - 4

22

3 xy 2

5

4 xy

3

1

3

1 xy

x + 3y = 1

3y = -x + 1

From the graph, the lines appear to intersect at (–5, –2).

-5y = -4x -10

4x – 5y = -10 2x – 7y = 4

-7y = -2x + 4

7

4

7

2 xy

Consistent & Independent Consistent & Independent

GUIDED PRACTICE

From the graph, the lines appear to intersect at (0, –8).

3. 8x – y = 8 3x + 2y = -16

8x – y = 8 3x + 2y = -16

-y = -8x + 8

y = 8x - 8

2y = -3x - 16

82

3 xy

Consistent & Independent

Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent.

4x – 3y = 8

8x – 6y = 16

4x – 3y = 8

– 3y = -4x + 8

8x – 6y = 16

– 6y = -8x + 16

(the equations are exactly the same)

2x + y = 4

2x + y = 1

the system has no solution

inconsistent.

2x + y = 4 2x + y = 1

(the lines have the same slope)

consistent and dependent.

The system has infinite solutions

3

8

3

4 xy

3

8

3

4 xy

y = -2x + 4 y = -2x + 1

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.

2x + 5y = 6

5y = -2x + 6

4x + 10y = 12

10y = -4x + 12 Same equationInfinite solutions

Consistent and independent

2x + 5y = 6

4x + 10y = 12A.

3x – 2y = 10

3x – 2y = 2B.

2y = – 3x + 10

3x – 2y = 10 Same slope // lines

no solutioninconsistent

3x – 2y = 2

2y = – 3x + 2

C. – 2x + y = 5

y = – x + 2

– 2x + y = 5

5

6

5

2 xy 5

6

5

2

xy

52

3 xy 1

2

3 xy

y = – x + 2y = 2x + 5

(–1, 3)

consistent

independent

C. – 2x + y = 5

y = – x + 2

– 2x + y = 5y = – x + 2

y = 2x + 5

(–1, 3)

consistent

independent

Is (-1,3) the correct solution?

– 2x + y = 5 y = – x + 2

– 2(-1) + (3)= 5

2 + 3 = 5

3 = – (-1) + 2

3 = 1 + 2 ☺☺

HOMEWORK 3.1P.156 #3-10 and board work

No solution

(3, 3)

Infinite solutions

(-1, 1)

5. 2

4

xy

xy

Solve each system of equations by graphing. Indicate whether the system is Consistent- Independent, Consistent-Dependent, or Inconsistent

6. 52

42

xy

xy

462

23

xy

xy42

2

xy

xy7. 8.

1022

6

yx

yx

1833

6

yx

yx9. 10.

xy

xy

82

421

02

32

yx

xy11. 12.

2

42

yx

yx

1222

2

yx

yx13. 14.

63

1

2

32

yx

xy

14

1

2

3

42

yx

yx15. 16.

(-1, 3)

Consistent, independent

5.

no solution

inconsistent,

6.

y = 3x - 2

Infinite solutions

Consistent, dependent

7.(1, 2)

Consistent, independent

8.

9.

y = x + 6y = x + 5

No solutions

Inconsistent

10.

y= -x + 6y = -x + 6

Infinite solutions

Consistent, dependent

11.

y = ½ x

(2, 1)

Consistent, independent

13.

y = -2x + 4y = x - 2

(2, 0)

Consistent, independent

14.

y = -x + 2y = -x + 6

No solutions

inconsistent15.

y = -3x + 2Consistent, dependent

Infinite solutions16.

y = -2x + 4y = 6x - 4

(1, 1)

Consistent, independent

12.

y = 1/2x + 4 Consistent, dependent

Infinite solutions