Topic 5A: Linear Equations Mrs. Daniel Algebra 1

Table of Contents Rate of Change & Slope 3 Forms of Linear Equations Slope-Intercept Form Point-Slope Form Standard Form

Rate of Change & Slope

Rate of Change change in dependent variable (y) rate of change = Rate 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. change in dependent variable (y) rate of change = change in independent variable (x)

Identify the Change of Rate 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?

Let’s Practice… 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?

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

Special Cases Find the slope of each line. A. B.

Let’s Practice… Tell whether the slope of each line is positive, negative, zero or undefined. A. B. The line rises from left to right. The line falls from left to right. The slope is positive. The slope is negative.

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.

Finding Slope Algebraically

Let’s Practice…. (3, 5), (2, 4) (-3, 1), (-2, 5) (8, 4), (6, -5)

3 Forms of Linear Equations

3 Forms for the Equation of a Lines Slope Intercept y = mx + b 2. Point Slope y – y 1 = m (x – x1) 3. Standard From Ax + By = C

3 Forms of Linear Equations

Finding the Equation of a Line Which formula you chose, depends on the information provided. You will use all three formulas to create linear equations

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: slope= 5, y-intercept = -7 slope= 2, y-intercept = -1 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 = -1 y = 2x - 1 3. slope= 3 and point (0, -2) y = 3x -2

Graphing Using Slope-Intercept Start at the y-intercept. Draw dot. Count slope in the positive direction. Draw dot. Count slope in the negative direction. Draw dot. Connect dots.

Let’s Practice… Graph: y = 2 3 𝑥−4

Let’s Practice… Graph: y = - 1 3 𝑥+2

Word Problems A 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 Form Use when given either: For example: 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. Slope = 2, passing through (3, 5) Slope = 4, passing through (1, 3)

Let’s Practice… y – 5 = 2(x - 3) y – 3 = 4 (x – 1) 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… Passing through (1, 2) and (5, 10) Passing through (3, 5) and (8, 15) Hint: Find the slope 1st

Let’s Practice… Slope : 𝟏𝟎 − 𝟐 𝟓 −𝟏 = 𝟖 𝟒 = 2 y – 2 = 2 (x – 1) Find the equation of the line… 1. Passing through (1, 2) and (5, 10) Slope : 𝟏𝟎 − 𝟐 𝟓 −𝟏 = 𝟖 𝟒 = 2 y – 2 = 2 (x – 1) 2. Passing through (3, 5) and (8, 15) Slope : 𝟏𝟓 −𝟓 𝟖 −𝟑 = 𝟏𝟎 𝟓 = 3 y – 5 = 3 (x – 3)

Graphing Using Point-Slope Identify the point. Graph. Identify the slope. Count slope in the positive direction. Draw dot. Count slope in the negative direction. Draw dot. Connect points.

Let’s Practice… Graph: y + 5 = -(x + 2)

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

Word Problem A 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: (3, 2) and (-1, 4) y-int = 3 and (-1, 2)

Mixed Practice… Determine the equation of the line. Write the final answer in slope-intercept format: 3. m = 2 3 and (0, -2) 4. (5, 2) and (3, 0)

Standard Form

Standard Form Ax + By = C

Finding Intercepts From a Graph Find the x- and y-intercepts.

Finding Intercepts Algebraically To find x-intercept, plug in zero for y and solve. To find y-intercept, plug in zero for x and solve.

Let’s Practice… Find the x and y intercepts. 5x – 6y = 60 3x + 8y = 12

Graphing Using Intercepts Determine x and y intercepts. ****You have to plug in zero, twice!!*** Plot points. X-intercept = (X, 0) Y-intercept = (0, Y) 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 = - 3 7 x + 5 to standard format.

Let’s Practice… y -2 = - 1 3 (x + 6) 2. y = - 2 3 𝑥 −1