Today:Warm-Up

Solutions to EquationsSlope of a Line

x-and y- InterceptsClass Work

Weekend

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TGIF: December 14, 2012

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Warm-Up

1. A 66 ft. board is cut into three pieces. The second piece is 1.5 times longer than the first. The third piece is twice as long as the second. How long is each piece?

2. Greg has $160 and plans to use some of the money for 5 basketball tickets. Write an expression showing the amount of money Greg has after buying the tickets.

3. Three quarts per minute equals how many gallons an hour?4. Solve and graph the following: -5< x - 4 <2

5. Solve and graph the following: -3h < 19 or 7h - 3> 18

Class Notes & Practice Problems Section of Notebook

Graphs of Linear Equations

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Graphs of y = bWhere b is any number

y = b are always horizontal lines, and are functions

Graphs of x = aWhere a is any number

x = a are always vertical lines, and are not functions

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Graphs of y = mxy = "a number times x"

y = mx lines will always go through the origin, and will be at the angle shown.

Graphs of y = mx + cy = "a number times x plus c"

y = mx + c are similar in shape to y = mx, but do not go through the origin.

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Defining Linear Equations

HOW DO I KNOW IF AN EQUATION IS LINEAR?

If an equation is linear, a constant change in the x-value corresponds to a

constant change in the y-value.

No bend or curve in the

line.

Not Linear

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Solutions to Linear Equations

To ask "Is the ordered pair (1,3) a solution to the equation y = 10x - 7?", is the same as asking, " Is the point (1,3) on the line of y = 10x -7"? How do we know?

Determine whether the following ordered pairs are solutions to the equation y = 10x -7a.) (1,3) b.) (2, 13) c.) (-1,-3)

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y

x2-2

The slope of a line is a number, m, which measures its steepness.

m = 0

m = 2m is undefined

m =1

2

m = -1

4

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x

y

x2 – x1

y2 – y1

change in y

change in x

The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula

The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).

y2 – y1

x2 – x1

m = , (x1 ≠ x2 ).

(x1, y1)

(x2, y2)

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Example: Find the slope of the line passing through the points (2, 3) and (4, 5).

Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5.

y2 – y1

x2 – x1

m = 5 – 3

4 – 2= =

22

= 1

2

2(2, 3)

(4, 5)

x

y

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Finding the x-and y-intercepts of Linear Equations

The path of the defender crosses the path of the thrown football.

In algebra, what are x- and y-intercepts?

What does it mean to INTERCEPT a pass in football?

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These are the x- and y-Intercepts

The x-intercept is where the graph crosses the x-axis. The y-coordinate is always zero.

The y-intercept is where the graph crosses the y-axis. The x-coordinate is always zero.

(2, 0)

(0, 6)

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x-intercept: Plug in 0 for y.

x - 2(0) = 12

x = 12; (12, 0)

y-intercept: Plug in 0 for x.

0 - 2y = 12

y = -6; (0, -6)

Find the x- and y-intercepts.1. x - 2y = 12

x-intercept: Plug in 0 for y.

-3x - 5(0) = 9

-3x = 9

x = -3; (-3, 0)

y-intercept: Plug in 0 for x.

-3(0) + 5y = 9

5y = 9

y = ; (0, )

Find the x- and y-intercepts.2. -3x + 5y = 9

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x-intercept: Plug in 0 for y.

Does 0 = 7?

No! There is no x-intercept. None

What type of lines have no x-intercept?

Horizontal!

Horizontal lines…y = 7…y-int = (0, 7)

Find the x- and y-intercepts.3. y = 7 ***Special case***

What is the x-intercept of3x – 4y = 24?

1. (3, 0)

2. (8, 0)

3. (0, -4)

4. (0, -6)

What is the y-intercept of-x + 2y = 8?

1. (-1, 0)

2. (-8, 0)

3. (0, 2)

4. (0, 4)

What is the y-intercept ofx = 3?

1. (3, 0)

2. (-3, 0)

3. (0, 3)

4. None

Class Work:8-3: 28-36;

Plot & Label 3 points per problem4.3-- All

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A linear equation written in the form y = mx + b is in slope-intercept form.

To graph an equation in slope-intercept form:

1. Write the equation in the form y = mx + b. Identify m and b.

The slope is m and the y-intercept is (0, b).

2. Plot the y-intercept (0, b).

3. Starting at the y-intercept, find another point on the line using the slope.

4. Draw the line through (0, b) and the point located using the slope.

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Example: Graph the line y = 2x – 4.

2. Plot the y-intercept, (0, -

4).

1. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4.

3. The slope is 2.

The point (1, -2) is also on the line.

1= change in y

change in xm = 2

4. Start at the point (0, 4). Count 1 unit to the right and 2 units up to locate a second point on the line.

2

x

y

5. Draw the line through (0, 4) and (1, -2).

(0, - 4)

(1, -2)

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A linear equation written in the form y – y1 = m(x – x1) is in point-slope form.

The graph of this equation is a line with slope m passing through the point (x1, y1).

Example:

The graph of the equation

y – 3 = - (x – 4) is a line

of slope m = - passing

through the point (4, 3).

1

2 1

2

(4, 3)

m = -1

2

x

y

4

4

8

8

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Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3.

Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1) = (-2, 5).

y – y1 = m(x – x1) Point-slope form

y – y1 = 3(x – x1) Let m = 3.

y – 5 = 3(x – (-2)) Let (x1, y1) = (-2, 5).

y – 5 = 3(x + 2) Simplify.

y = 3x + 11 Slope-intercept form

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Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5).

y – y1 = m(x – x1) Point-slope form

Slope-intercept formy = - x + 13

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3

2 1 5 – 3 -2 – 4

= - 6

= - 3

Calculate the slope.m =

Use m = - and the point (4, 3).y – 3 = - (x – 4)1

3 3

1

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Two lines are parallel if they have the same slope.

If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2.

Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m1 = 2 and m2 = 2.

The lines are parallel.

x

y

y = 2x + 4

(0, 4)

y = 2x – 3

(0, -3)

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Two lines are perpendicular if their slopes are negative reciprocals of each other.If two lines have slopes m1 and m2, then the lines

are perpendicular whenever

The lines are perpendicular.

1m1

m2= - or m1m2 = -1. y = 3x – 1

x

y

(0, 4)

(0, -1)

y = - x + 41

3Example:

The lines y = 3x – 1 and

y = - x + 4 have slopes

m1 = 3 and m2 = - .

1

3 1

3

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Equations of the form ax + by = c are called linear equations in two variables.

The point (0,4) is the y-intercept.

The point (6,0) is the x-intercept.

(0,4)

(6,0)x

y

2-2

This is the graph of the equation 2x + 3y = 12.