6.1 Solving systems by graphing

Solving systems by graphing and analyzing special systems

Solution: An ordered pair that makes the equation true.

Example:x + y = 7

What are some possible solutions?

5 + 2 = 7

3 + 4 = 7

-1 + 8 = 7

-10 + 17 = 7

(5,2)

(3,4)

(-1,8)

(-10,17)

The solutions are always written in ordered pairs.:

How many more solutions are there?

System of EquationsDefinition: a set of two or more equations that

have variables in common.

Example ofa system:

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

Solution: Any ordered pair that makes ALL the equations in a system true.

Solution for this system: (-1,-2)

( ) = -2( ) - 4 -2 = 2 - 4

Verify (check your answer) whether or not the given ordered pair is a solution for the following system of equations. (same system as previous slide)

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

( -1,-2)

( ) = 3 ( ) + 1 -1

-1

-2

-2

-2 = -3 + 1

Is (-1, -2) a solution? explain

YES, because the ordered pair makes both equations true.

-5

0

5

-5 0 5

Find the solution for the following system.

y = 3x + 1

y = -2x - 4

m= 3; b = 1

m= -2; b = -4

Solution: (-1, -2)

-5

0

5

-5 0 5

Find the solution for the following system.

2x + y = 3 y = -2x + 3

y = 3x - 2

m= -2; b = 3

m= 3; b = -2

Solution: (1, 1)

-5

0

5

-5 0 5

Find the solution for the following system.

y = ½ x + 1

2y – x = 2

m= ½; b = 1

x –int=

y –int=

-2

1

Solution:Infinitely many solutions (because the lines are the same)

-5

0

5

-5 0 5

Find the solution for the following system.

y = 2x + 2

y = 2x - 1

m= 2; b = 2

m= 2; b = -1

Solution:No solution (because the lines are parallel)

Writing a system of equations.

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 alligators weigh the same amount?

Alligator 1Initial weight: 4 poundsRate of growth: 1.5 lb/ month

Alligator 2Initial weight: 6 poundsRate of growth: 1 lb/ month

Alligator weight (w) = ___________ + ________________________ Initial weight Growth rate times Time (t)

Alligator 1: w = 4 + 1.5t

Alligator 2: w = 6 + 1 t

Slope = w-intercept = Slope = w-intercept =

1.51

46

Alligator weights121086420

WEIGHT

0 1 2 3 4 5 6time

Answer?

4 months

Systems with infinitely many solutions or no solutions.

Slopes: differentIntersect: one point

Slopes: samey-intercept: same

The lines are the same

Slopes: sameLines are parallel

lines don’t intersect

Assignment: pg 367: 10-38 evensScales other than 1: 10,12,16?