Section 5-2

Solving Systems Using Tables or Graphs

Warm-up

Graph the following equations on the same graph.

a. y = 2/3x - 5

b. y = -x + 3

Warm-up

Graph the following equations on the same graph.

a. y = 2/3x - 5

b. y = -x + 3x

y

y = 2/3x - 5

y = -x + 3

How can we solve a system?

System:

System: Two sentences connected by “and”

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

Notation:

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

Notation:

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

Notation:

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

Solution Set of a System:

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

Notation:

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

Solution Set of a System: Since we’re using “and,” we are looking for what is in both sets/equations

System: Two sentences connected by “and”

y = 4x + 8 and y = 3x + 5

Notation:

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

Solution Set of a System: Since we’re using “and,” we are looking for what is in both sets/equations

**Often, it is just a point**

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

0

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

0 8 5

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

0 8 5 3

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01

8 5 3

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01

8 512 8

3

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01

8 512 8

34

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2

8 512 8

34

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2

8 512 80 -1

34

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2

8 512 80 -1

341

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2-3

8 512 80 -1

341

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2-3

8 512 80 -1-4 -4

341

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2-3

8 512 80 -1-4 -4

3410

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2-3

8 512 80 -1-4 -4

3410

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

a. Tables x y = 4x + 8 y = 3x + 5

01-2-3

8 512 80 -1-4 -4

3410

(x, y) = (-3, 4)

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc3. Choose 5: intersect

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc3. Choose 5: intersect

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc3. Choose 5: intersect

4. Choose the two equations

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc3. Choose 5: intersect

4. Choose the two equationsx

y

Example 1

Solve the system

y = 4x + 8

y = 3x + 5

⎧⎨⎩⎪

b. Graphing (by hand)

c. Graphing Calculator

1. Graph both equations

x

y

2. Press 2nd --> Calc3. Choose 5: intersect

4. Choose the two equationsx

y

(x, y) = (-3, 4)

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

w

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

l + 2w = 70

lw = 400

⎧⎨⎩⎪

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

l + 2w = 70

lw = 400

⎧⎨⎩⎪

Can we accurately solve this by graphing?

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

l + 2w = 70

lw = 400

⎧⎨⎩⎪

Can we accurately solve this by graphing?

(l, w) ≈ (14.4, 27.8)

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

l + 2w = 70

lw = 400

⎧⎨⎩⎪

Can we accurately solve this by graphing?

(l, w) ≈ (14.4, 27.8)(l, w) ≈ (55.6, 7.2)

Example 2

Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do

this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

Barn l

w

ww = width, l = length

l + 2w = 70

lw = 400

⎧⎨⎩⎪

Can we accurately solve this by graphing?

(l, w) ≈ (14.4, 27.8)(l, w) ≈ (55.6, 7.2)

14.4m x 27.8m or55.6m x 7.2 m

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

a. Table

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

0

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

0 0 1

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

0 0 1 1

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01

0 1 1

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01

0 15 -2

1

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01

0 15 -2

17

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1

0 15 -2

17

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1

0 15 -2-5 4

17

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1

0 15 -2-5 4

17-9

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1

0 15 -2-5 4

17-9

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1

0 15 -2-5 4.5 .7

17-9

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1

0 15 -2-5 4.5 .7

17-9.2

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1.2

0 15 -2-5 4.5 .7

17-9.2

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1.2

0 15 -2-5 4.5 .71 .4

17-9.2

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1.2

0 15 -2-5 4.5 .71 .4

17-9.2.6

Example 3

Solve

y = 5x

y = −3x +1

⎧⎨⎩⎪

x y = 5x y = -3x + 1a. Table

01-1.1.2

0 15 -2-5 4.5 .71 .4

17-9.2.6

How long do we try this?

Example 3

Solve

b. Graphing (by hand)

y = 5x

y = −3x +1

⎧⎨⎩⎪

Example 3

Solve

b. Graphing (by hand)

y = 5x

y = −3x +1

⎧⎨⎩⎪

c. Graphing Calculator

Example 3

Solve

b. Graphing (by hand)

y = 5x

y = −3x +1

⎧⎨⎩⎪

c. Graphing Calculatorx

y

Example 3

Solve

b. Graphing (by hand)

y = 5x

y = −3x +1

⎧⎨⎩⎪

c. Graphing Calculatorx

y

x

y

Example 3

Solve

b. Graphing (by hand)

y = 5x

y = −3x +1

⎧⎨⎩⎪

c. Graphing Calculatorx

y

x

y(x, y) = (.125, .625)

Homework

Homework

p. 282 #1-24

“Lack of money is no obstacle. Lack of an idea is an obstacle.” - Ken Hakuta