LESSON 4.1 – WRITING EQUATIONS IN SLOPE-INTERCEPT FORM Learning Goal: Students will be able to write equations in slope- intercept form and use linear equations to solve real-life problems.

REMEMBER- Slope-Intercept form: y = mx + b where m is the slope and b is the y-intercept

EXAMPLE 1A Write an equation of each line with the given slope and y- intercept. Slope = -3; y-intercept = ½ Step 1: write the slope-intercept form y = mx + b Step 2: substitute -3 for m and ½ for by = -3x + ½ The equation is y = -3x + ½

EXAMPLE 1B Write an equation of each line with the given slope and y- intercept. Slope = 0; y-intercept = -2 Step 1: write the slope-intercept form y = mx + b Step 2: substitute 0 for m and -2 for b y = (0)x + (-2) Step 3: simplifyy = -2 The equation is y = -2

Find the slope and y-intercept Let (x 1, y 1 ) = (0, -3) and (x 2, y 2 ) = (4, 3) m = Because the line crosses the y axis at (0, -3) the y- intercept is -3. So the equation is y = 3x – 3 2 y 2 -y 1 3-(-3) 6, or 3 x 2 -x EXAMPLE 2A = =

Find the slope and y-intercept Let (x 1, y 1 ) = (0, 2) and (x 2, y 2 ) = (4, -1) m = Because the line crosses the y axis at (0, 2) the y- intercept is 2. So the equation is y = -3x y 2 -y x 2 -x EXAMPLE 2B = =

EXAMPLE 3A Write an equation of the line that passes through the given points. (-3, 5), (0, -1) You need to find the slope and the y-intercept. m = Because the line crosses the y-axis at (0, -1), the y-intercept is -1. So, the equation is y = -2x – (-3) = -2

EXAMPLE 3B Write an equation of the line that passes through the given points. (0, -5), (8, -5) You need to find the slope and the y-intercept. m = Because the line crosses the y-axis at (0, -5), the y-intercept is -5. So, the equation is y = (0)x – 5 or y = – (-5) 8 – 0 = 0

EXAMPLE 4 Write a linear function f with the values of f(0) = 10 and f(6) = 34. Step 1: Write the values as coordinates f(0) = 10 is (0,10) f(6) = 34 is (6, 34) Step 2: Find the slope that passes through the points (0,10) and (6,34) m Step 3: Find the y- intercept Because the line crosses the y-axis at (0,10) the y-intercept is 10. Step 4: Write the equation f(x) = 4x – – 0 6 = = = 4

YOU TRY Write an equation of the line with the given slope and y-intercept. 1.Slope = 7, y-intercept = 2 2.Slope = 1/3, y-intercept = -1 Write an equation of the line in slope-intercept form y = 7x + 2 y = 1/3x – 1 y = 1/2x + 1 y = -2/5x – 1

YOU TRY 5. Write an equation of the line that passes through (0, -2) and (4, 10). 6. Write a linear function g with the values g(0) = 9 and g(8) = 7. y = 3x -2 y = -1/4x + 9

SOLVING REAL LIFE PROBLEMS A linear model is a linear function that models a real-life situation. When a quantity y changes at a constant rate with respect to a quantity x, you can use the equation y = mx + b to model the relationship. The value of m is the constant rate of change, and the value of b is the initial, or starting, value of y.

EXAMPLE 5 Excluding hydropower, U.S. power plants used renewable energy sources to generate 105 million megawatt hours of electricity in By 2012, the amount of electricity generated had increased to 219 million megawatt hours. Write a linear model that represents the number of megawatt hours generated by non-hydropower renewable energy sources as a function of the number of years since Use the model to predict the number of megawatt hours that will be generated in 2017.

1. Understand the problem You know the amounts of electricity generated in two distinct years. You are asked to write a linear model that represents the amount of electricity generated each year since 2007 and then predict a future amount. 2. Make a Plan Break the problem into parts and solve each part. Then combine the results to help you solve the original problem.  Part 1 Define the variables. Find the initial value and the rate of change.  Part 2 Write a linear model and predict the amount in 2017.

3. Solve the problem  Part 1Let x represent the time (in years since 2007) Let y represent the number of megawatt hours (in millions). Since x is the years since 2007, 2007 corresponds to x = corresponds to x = Let (x 1, y 1 ) = (0, 105) and (x 2, y 2 ) = (5, 219). m = The initial value is the y-intercept b, which is y 2 – y – X 2 – x 1 5 – 0 5 = == 22.8

3. Solve the problem  Part 2m = 22.8; b = 105 Write the equation in the format y = mx + b. y = 22.8x Megawatt hours (millions) Rate of change Years since 2007 Initial Value =. + Predict the amount in Corresponds to x = Substitute 10 for x and solve for y. y = 22.8 (10) Y = Y = 333 In 2017, non-hydropower renewable energy sources will generate 333 million megawatt hours. 10

YOU TRY 7. The corresponding data for electricity generated by hydropower are 248 million megawatt hours in 2007 and 277 million megawatt hours in Write a liner model that represents the number of megawatt hours generated by hydropower as a function of the number of years since y = 5.8x + 248

CLOSURE For a line that has been graphed in a coordinate plane, you can write the equation by…