Section 4.5: Graphing Linear Equations

ObjectivesThe student will be able to:

EA 4.7- 1. graph linear functions.

2. write equations in standard form.

Solving for Linear Equations

• In order for us to solve and GRAPH linear equations, we need to review solving linear equations (from chapter 3)…

• Each (most) problems will have two variables, x and y.

• You will always solve for y! Remember we must undo to solve for y… let’s review.

Graphing Steps

1) Isolate the variable (solve for y).

2) Make a t-table. If the domain is not given, pick your own values:

(-2, -1, 0, 1, 2) for x values.

3) Plot the points on a graph.

4) Connect the points.

1) Review: Solve for y 2x + y = 4

1. Draw “the river”

2. Subtract 2x from both sides

- 2x - 2x

y = -2x + 42) Solve for y: 4x + 2y = -6

1. Subtract 4x

2. Simplify

3. Divide both sides by 2

4. Simplify

- 4x - 4x

2y = -4x - 6

2 2

y = -2x - 3

3) Solve for y: x - 3y = 6

1. Subtract x

2. Simplify

3. Divide both sides by -3

4. Simplify

- x - x

-3y = -x + 6

-3 -36

3

xy

2

3

xy

or

4) Review: Make a t-tableIf f(x) = 2x + 4, complete a table using the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)

2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)

2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)-2

-1

0

1

2

ordered pair

5) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)

-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-1

0

1

2

Bonus questions!What is the x-intercept?

(2, 0)What is the y-intercept?

(0, 6)Does the line increase or decrease?

Decrease

6) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 63. Plot the points

(-2,12), (-1,9), (0,6), (1,3), (2,0)

4. Connect the points.

A. .

B. .

C. .

D. .

7. Which is the graph of y = x – 4?

Standard FormAx + By = C

A, B, and C have to be integers

An equation is LINEAR (the graph is a straight line) if it can be written in standard form.

This form is useful for graphing (later on…).

Linear Equations

• To be LINEAR:

• No exponents

• No variables multiplied together Ex. xy = 4

• No variables in the denominator (bottom)

• CAN have y = #, Example: y = 4

• Look at the next slide for examples =>

Here’s the cheat sheet! An equation that is linear does NOT contain the following:

1. Variables in the denominator

2. Variables with exponents

3. Variables multiplied with other variables.

xy = 12

32y

x

2 3y x

8) Determine whether each equation is a linear equation.

1) 4x = 7 + 2y

Can you write this in the form

Ax + By = C?

4x - 2y = 7

A = 4, B = -2, C = 7

This is linear!

2) 2x2 - y = 7Can you write it in standard form?

NO - it has an exponent!Not linear

3) x = 12x + 0y = 12

A = 1, B = 0, C = 12Linear

9. Determine whether each equation is a linear equation.

10. Is this equation linear?

1. Yes

2. No

4 3x y

Standard Formx – 4y = 3

11. Is this equation linear?

A. Yes

B. No

29 4y x

Exponents are not allowed!

12. Is this equation linear?y = -3

A. Yes

B. No

Standard Form0x + y = -3

Warm Up, copy the Agenda

• Solve the following for y:

• 1. x = 4y

• 2. 2x + 3y = 44

• 3. 2x + y = 104

• 4. x – 2y = 2

• 5. x – y = -3

• 6. x – y = 10

1) Review: Solve for y 2x + y = 4

1. Draw “the river”

2. Subtract 2x from both sides

- 2x - 2x

y = -2x + 42) Solve for y: 4x + 2y = -6

1. Subtract 4x

2. Simplify

3. Divide both sides by 2

4. Simplify

- 4x - 4x

2y = -4x - 6

2 2

y = -2x - 3

3) Solve for y: x - 3y = 6

1. Subtract x

2. Simplify

3. Divide both sides by -3

4. Simplify

- x - x

-3y = -x + 6

-3 -36

3

xy

2

3

xy

or

A. .

B. .

C. .

D. .

4. Which is the graph of y = x – 4?

5) Review: Make a t-tableIf f(x) = 2x + 4, complete a table using the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)

2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)

2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)-2

-1

0

1

2

ordered pair

Steps:

• So first step is to SOLVE FOR Y

• Plug in the x values (to get y)

• Label points (x, y)

• Graph either points or double check with graphing calculator!

• You are done.

6) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)

-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-1

0

1

2