topic 5a: linear equations mrs. daniel algebra 1

Topic 5A:Linear Equations

Mrs. DanielAlgebra 1

Table of Contents1. Rate of Change & Slope2. 3 Forms of Linear Equations3. Slope-Intercept Form4. Point-Slope Form5. Standard Form

Rate of Change & Slope

Rate of ChangeRate of Change – a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

The rates of change for a set of data may vary or be constant.

The table shows the average temperature (°F) for five months in Chicago. Find the rate of change for each time period. During which time period did the temperature increase at the fastest rate?

Identify the Change of Rate

The table shows the balance of a bank account on different days of the month. Find the rate of change during each time interval. During which time interval did the balance decrease at the greatest rate?

Let’s Practice…

Rates of Change Graphically

What is Slope?• Slope: describes the steepness or incline of a

line. A higher slope value indicates a steeper incline.

• Slopes can be positive, negative, zero or undefined.

• Slope is abbreviated with “m”

Determining Slope Graphically• We can

count the rise and run on a graph to determine slope.

Forms of Slopes

Find the slope of each line.A. B.

Special Cases

Tell whether the slope of each line is positive, negative, zero or undefined.

The line rises from left to right. The line falls from left to right.

The slope is positive. The slope is negative.

A. B.

Let’s Practice…

Find the slope of the line that contains (0, –3) and (5, –5) graphically.

Let’s Practice…

Find the slope of the line graphically.Let’s Practice…

Finding Slope Algebraically

slope  = 𝒚 𝟐− 𝒚𝟏𝒙 𝟐− 𝒙𝟏

Let’s Practice….1. (3, 5), (2, 4)

2. (-3, 1), (-2, 5)

3. (8, 4), (6, -5)

3 Forms of Linear Equations

3 Forms for the Equation of a Lines

1. Slope Intercepty = mx + b

2. Point Slopey – y 1 = m (x – x1)

3. Standard FromAx + By = C

3 Forms of Linear Equations

• Which formula you chose, depends on the information provided.

• You will use all three formulas to create linear equations

Finding the Equation of a Line

Slope Intercept

Form

Slope-Intercept From Use when given:

–Slope and y-intercept–Slope and point (0, ??)

For example:–What is the equation of line with a slope

of 3 and y-intercept of 6?

What is the y-intercept?• The y-intercept of a line is the point at which

the line crosses the y axis. It is where the x value equals 0.

• y-intercept = ( 0, y )

Let’s Practice….Find the equation of the line:1. slope= 5, y-intercept = -7

2. slope= 2, y-intercept = -1

3. slope= 3 and point (0, -2)

Let’s Practice….Find the equation of the line:1. slope= 5, y-intercept = -7

y = 5x -7

2. slope= 2, y-intercept = -1y = 2x - 1

3. slope= 3 and point (0, -2)y = 3x -2

Graphing Using Slope-Intercept

1. Start at the y-intercept. Draw dot.

2. Count slope in the positive direction. Draw dot.

3. Count slope in the negative direction. Draw dot.

4. Connect dots.

Let’s Practice…Graph:y =

Let’s Practice…Graph:y = -

Word ProblemsA carpenter charges a $45 fee plus $30 per hour for labor. Write an equation to model the total cost of a job. Draw a graph models the total cost.

Point-Slope Form

Point-Slope FormUse when given either:

– A point and the slope– 2 points

For example:– Find the equation for a line with points (3, 2) and

a slope of -4.

Let’s Practice…Find the equation of the line in point-slope.1. Slope = 2, passing through (3, 5)

2. Slope = 4, passing through (1, 3)

Let’s Practice…Find the equation of the line in point-slope.1. Slope = 2, passing through (3, 5)

y – 5 = 2(x - 3)

2. Slope = 4, passing through (1, 3)

y – 3 = 4 (x – 1)

Let’s Practice…Find the equation of the line…1. Passing through (1, 2) and (5, 10)

2. Passing through (3, 5) and (8, 15)

Hint: Find the slope 1st

Let’s Practice…Find the equation of the line…1. Passing through (1, 2) and (5, 10)

Slope : = = 2 y – 2 = 2 (x – 1)

2. Passing through (3, 5) and (8, 15)

Slope : = = 3 y – 5 = 3 (x – 3)

Graphing Using Point-Slope1. Identify the point. Graph.2. Identify the slope. 3. Count slope in the positive direction. Draw

dot.4. Count slope in the negative direction. Draw

dot.5. Connect points.

Let’s Practice…Graph:y + 5 = -(x + 2)

Let’s Practice…Graph:y - 4 = -2(x + 1)

Word ProblemA restaurant’s goal is to serve 600 customers in 8 hours and 900 customers in 12 hours. Write an equation in point-slope form that represents the number of customers served per hour. What is the graph of the equation?

Mixed Practice…Determine the equation of the line. Write the final answer in slope-intercept format:

1. (3, 2) and (-1, 4)

2. y-int = 3 and (-1, 2)

Mixed Practice…Determine the equation of the line. Write the final answer in slope-intercept format:3. m = and (0, -2)

4. (5, 2) and (3, 0)

Standard Form

Standard Form

Ax + By = C

Find the x- and y-intercepts.

Finding Intercepts From a Graph

Finding Intercepts Algebraically

1.To find x-intercept, plug in zero for y and solve.

2.To find y-intercept, plug in zero for x and solve.

Let’s Practice…Find the x and y intercepts. 1. 5x – 6y = 60

2. 3x + 8y = 12

Graphing Using Intercepts1. Determine x and y intercepts.

****You have to plug in zero, twice!!***

2. Plot points.X-intercept = (X, 0)Y-intercept = (0, Y)

3. Connect dots.

Let’s Practice…Graph:• x – 2y = -2

Let’s Practice…Graph:• 2x + 5y = 20

Graphing Horizontal Lines

Graphing Vertical Lines

Transforming to Standard Form

Use algebra to rearrange variables to desired format.

Example: Transform: y = - x + 5 to standard format.

Let’s Practice…1. y -2 = - (x + 6)

2. y = -