Writing Linear Equations in Slope-Intercept Form

Video Links www.youtube.com/watch?v=xEjK1oasLSY www.youtube.com/watch?v=LTb2-LE7StE www.youtube.com/watch?v=gf-829r4B4s www.youtube.com/watch?v=LoeMbCdkVhY

Writing Linear Equations given slope and y-intercept Put the equation of a line in slope-intercept form y = mx + b where m is the slope and b is the y- intercept. The y-intercept is where a line crosses the y- axis.

Writing an Equation of a Line given slope and y-intercept Suppose the slope of a line is 5 and the y- intercept is 2. How would this you write the equation of this line in slope-intercept form? First write the slope-intercept form. y = mx + b Now substitute 5 for m and 2 for b. y = 5x + 2

Find the Equation of a Line Given the Graph Find the y-intercept from the graph. Count the slope from the graph. To write the equation of the line, substitute the slope and y-intercept in the slope-intercept form of the equation.

Write the Equation of a Line from a Graph Where does the line cross the y-axis? At the point (0, -4) The y-intercept is -4. What is the slope of the line? The graph also crosses the x-axis at (2, 0). We can use the slope formula to find our slope. m = -4 – 0 = -4 = 2 0 – 2 -2 We know our slope is 2 and our y-intercept is -4, what is the equation of our line? y = mx + b y = 2x + (-4) y = 2x -4

Let’s try some examples! Write the equation of a line with a slope of -2 and a y-intercept of 6. y = mx + b y = -2x + 6 Write the equation of a line with a slope of -4/3 and a y-intercept of 1. y = mx + b y = (-4/3)x + 1

Using a graph Where does the line cross the y-axis? At the point (0, 2) So the y-intercept b is 2. The line also passes through the point (3, 0). We can use these points to find the slope of the line. How? What formula do we use? Using the slope formula, we find that the slope m is -2/3. Write the equation of the line. y= mx + b y = (-2/3)x + 2

Example 1 x y +2 +3 b = -3 m = y = x - 3

Example 2 b = 1 m = y = x + 1 x y +1 -2

Given in standard form Put in slope intercept form 5x - 3y = 6 Write it in slope-intercept form. (y = mx + b) 5x – 3y = 6 -3y = -5x + 6 y = x - 2 m = b = -2 -3 -3 -3

Put in slope intercept form. 2y + 2 = 4x Write it in slope-intercept form. (y = mx + b) 2y + 2 = 4x 2y = 4x - 2 y = 2x - 1 m = 2 b = -1 2 2 2

If we are given the Slope and a Point Step 1: First find the y-intercept. Substitute the slope m and the coordinates of the given point (x, y) into the slope-intercept form, y = mx + b. Then solve for the y-intercept b. Step 2: Then write the equation of the line. Substitute the slope m and the y-intercept b into the slope- intercept form, y = mx + b.

Given the Slope and a Point Suppose we have a slope of -3 and it passes through the point (1, 2). We first need to find the y-intercept. We can do this by substituting our information into slope-intercept form and solving for b. y = mx + b 2 = -3(1) + b 2 = -3 + b Add 3 to both sides. 5 = b Now we know that the y-intercept is 5. y = -3x + 5

Try These! Suppose we have a line with a slope of -1 and passes through the point (3, 4). y = mx + b 4 = (-1)3 + b 4 = -3 + b 7 = b y = (-1)x + 7 y = -x + 7 Suppose we have a line with a slope of 2 and passes through the point (1, 3). y = mx + b 3 = 2(1) + b 3 = 2 + b 1 = b y = 2x + 1

Writing Equations given two points Calculate the slope (m) of the two points Use one of the point (x,y) and the slope (m) to find the y-intercept (b) Write the equation by substituting the slope and the y-intercept into the slope-intercept formula y = mx + b.

Writing Equations given two points Example: Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) and y – intercept (b) We need both!! First, we have to find the slope. Plug the points into the slope formula. Simplify

Writing Equations given two points Write an equation of the line that goes through the points (-2, 1) and (4, 2). slope (m) = y – intercept (b) = Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? (4, 2) because both numbers are positive. 2 = (4) + b ???

Writing Equations given two points 2 = (4) + b Solve the equation for b 2 = + b To write an equation, you need two things: slope (m) = y – intercept (b) =

Slope As shown in the previous examples, slope can be positive, negative, zero or undefined. You can tell which of these is the case by looking at a graph of a line–you do not need to calculate the slope.

Writing Equations of Parallel Lines Find the slope (m) of the two points Two lines are parallel if and only if they have the same slope Use one of the point (x,y) and the same slope (m) to find the y-intercept (b) Write the equation by substituting the slope and the y-intercept into the slope-intercept formula y = mx + b.

Writing Equations of Parallel Lines Example: Write the equation of a line that is parallel to the line y = 4x -3 and passes through the point (3, 2). Since the two lines are parallel then both lines have a slope of m = 4. We must substitute the slope and coordinates into the slope-intercept form and solve for b. 2 = 4(3) + b 2 = 12 + b Subtract 12 from both sides -10 = b Now we have enough information to write the equation of the line. y = mx + b y = 4x + (-10) y = 4x -10

Almost done! Write the equation of a line that is parallel to the line y = -2x -3 and passes through the point ( -2, 3). Since they are parallel, they both have the same slope m of -2. Now substitute our slope and coordinates into slope-intercept form. 3 = -2(-2) + b 3 = 4 + b -1 = b Now we can write the equation of the second line. y = mx + b y = -2x -1 Write the equation of a line that is parallel to the line y = 3x + 2 and passes through the point (4, -1). Since they are parallel, they both have the same slope m of 3. Now substitute our slope and coordinates into slope-intercept form. -1 = 3(4) + b -1 = 12 + b -13 = b Now we can write the equation of the second line. y = mx + b y = 3x -13

Writing Equations of Perpendicular Lines Find the slope (m) of the two points Two lines are perpendicular if and only if they are opposite reciprocal slopes flip and change sign Use one of the point (x,y) and the perpendicular slope (m) to find the y-intercept (b) Write the equation by substituting the slope and the y-intercept into the slope-intercept formula y = mx + b.

Writing Equations of Perpendicular Lines Example: Find the equation in slope-intercept form of the line that contains (1, 8) and is perpendicular to m = 8 = (-4/3)(1) + b 8 = -4/3 + b (add 4/3 to both sides) 28/3 = b

Using linear equation in a word problem. As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. Write the linear equation to model this situation.

Using linear equation in a word problem. her sales are $100, she makes $58 (100,58). When her sales are $300, she makes $78 (300, 78). Find slope 78 – 58 = 20 = 0.1 300 – 100 200 What is her daily Salary? (if she sales nothing)

Using linear equation in a word problem. What is her daily Salary? (if she sales nothing) What is her commission rate? If she sales $100, she makes $58 The slope of the line is 0.1 58 = 0.1(100) + b 58 = 10 + b -10 -10 48 = b y = 0.1x + 48

y = 0.1x + 48 So she earns $48 for just being there. Commission rate is 0.1 or 10% What would she earn if she sold $500?

y = 0.1x + 48 So she earns $48 for just being there. Commission rate is 0.1 or 10% What would she earn if she sold $500? y =0.1(500) + 48 y = 50 + 48 y = $98

to Graph the Equation of a Line Only two points are needed 1) Plot the y-intercept(b) on the y-axis. 2) From the y-intercept count using the slope to plot the next point. 3) Connect the two points