A linear equation in two variables is expressed in standard form Ax + By = C,

1. Today we will explore the Essential Question, “What is the method for graphing a linear equation in standard form form using the slope. the y-intercept and the x-intercepts?” A linear equation in two variables is expressed in standard form Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. If the equation is expressed in this form, the slope is , y-intercept when x = 0 is and the x-intercept when y = 0 is . Standard form of an Equation of a Line: Ax + By = C Where slope = , y-intercept = , x-intercept = .

2. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 The y-intercept of a line is the point where the line intersects the y -axis. Every point on the y-axis has an x-coordinate of 0. For example, if the y -intercept of a line is 3, then the line intersects the y -axis at the point (0,3). The line shown below has a y -intercept of 3.

3. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 The x-intercept of a line is the point where the line intersects the x -axis. Every point on the x-axis has a y-coordinate of 0. For example, if the x -intercept of a line is 1, then the line intersects the x -axis at the point (1,0). The line shown below has a x-intercept of 1. (1, 0)

4. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 Given the linear equation in standard form 2x – 3y = 12, find the slope, y- intercept and x-intercept. Also graph the function. The slope = , then , the y-intercept is ,then , and the x-intercept = , then Lets use the x-intercept and y-intercept to graph the linear equation. The x-intercept is (6, 0) and the y-intercept is (0, -4) If the equation is expressed in this form, the slope is , y-intercept when x = 0 is and the x-intercept when y = 0 is . (6, 0) (0, -4)

5. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 The line shown in the graph has a y-intercept of (0, 8) and a x-intercept of (-4, 0) Once we have the two intercepts lets find the slope of the line. Since we have two points lets use the slope formula. m = = Recall that the slope-intercept form of the equation of a line is where m denotes the slope and b denotes the y-intercept. We know that m = 2 and b = 8. 2 8 Substituting these values into the formula, we get the equation of the line Now we are going to write the equation in standard form. Lets get 8 by itself now , we can multiple both sides by -1, we get NOTE: There is an interactive website for graphing a line using the slope and the y-intercept at http://www.shodor.org/interactivate/activities/SlopeSlider/.

6. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 Modeled Examples: Example 1: Use the slope and the y-intercept and x-intercept to write an equation in standard form which could represent the graph of the line shown in the diagram. Solution: The x-intercept is (5, 0) and the y-intercept (0, 4) Since we have two points lets use the slope formula. m = = Substituting these into the formula y = mx + b, we get the equation of the line . Now we are going to write the equation in standard form. Lets get 4 by itself now, we can multiple both sides by 5, we get . Now we have to isolate the 20. we will add 4x to both sides,

7. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 Example 2. Given the linear equation in standard form 6x – 5y = 30, find the slope, y- intercept and x-intercept. Also graph the function. The slope = , then , the y-intercept is ,then , and the x-intercept = , then If the equation is expressed in this form, the slope is , y-intercept when x = 0 is and the x-intercept when y = 0 is . Lets use the x-intercept and y-intercept to graph the linear equation. The x-intercept is (5, 0) and the y-intercept is (0, -6) (5, 0) (0, -6)

8. y 10 8 6 4 x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -4 -6 -8 -10 Guided Practice Problems: (0, 8) 1. Roger went to a garage sale where hardback books sold for $5 each and paperback books sold for $2.50 each. He has $20 to spend. The equation below can be used to find how many books of each type Roger can buy, where x is the number of hardback books and y is the number of paperback books. 5x + 2.5y = 20 The slope = , then , the y-intercept is ,then , and the x-intercept = , then (4, 0) If the equation is expressed in this form, the slope is , y-intercept when x = 0 is and the x-intercept when y = 0 is . Find the x-intercept , y-intercepts and slope, then graph the equation

9. 1 2 -4 -3 -2 -1 1 2 3 4 Example 2. An engineer needs to determine the slope between two points on a gondola ride in order to evaluate the power requirements when the gondola is full of passengers. A coordinate grid has been placed over a diagram between the two points, as shown below. For estimation purposes, a straight line between the two points can be used to find the slope.

10. y 5 4 3 2 x 1 -5 -4-3 -2-10 1 234 5 -1 -2 -3 -4 -5 3. Use the slope and the y-intercept to graph the line whose equation is . Solution: The equation of the line is written in the form as shown, so and . Since b denotes the y-intercept and , the line passes through the point . Plot that point on the graph. We can use the definition of slope to find another point on the line. How can -3 be written as a ratio? First we must write the slope, m, as a ratio. We can use or any other ratio which denotes -3. Begin at the y-intercept and move down 3 and to the right 1. Plot that point. Draw the line through the two points plotted. This line is the graph of the given equation.

11. y 5 4 3 2 x 1 -5 -4-3 -2-10 1 234 5 -1 -2 -3 -4 -5 4. Use the slope and the y-intercept to graph the line whose equation is . Solution: The equation of the line is written in the form as shown, so and . Since b denotes the y-intercept and , the line passes through the point . Plot that point on the graph. We can use the definition of slope to find another point on the line. This slope, m, is already written as a ratio. Begin at the y-intercept and move up 2 and to the right 3. Plot that point. Draw the line through the two points plotted. This line is the graph of the given equation.

12. y y 5 x -10 -8-6-4 -2 0 2 4 6 8 10 4 3 2 x 1 -5 -4-3 -2-10 1 234 5 -1 -2 -3 -4 -5 10 8 6 4 2 -2 -4 -6 -8 -10 Independent Practice Problems: 1. Use the slope and the y-intercept to graph the line whose equation is . 2. Write the equation of the line whose graph is shown in slope-intercept form. +3 +2