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**Grade 8 Algebra1 The Slope Formula**

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**Tell whether the given ordered pairs satisfy a linear function.**

Warm Up Tell whether the given ordered pairs satisfy a linear function. 1) {(1, 1) , (2, 4) , (3, 9) , (4, 16)} 2) {(9, 0), (8, -5), (5, -20), (3, -30)} CONFIDENTIAL

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**The Slope Formula In the previous lesson, slope was described as**

the constant rate of change of a line. You saw how to find the slope of a line by using its graph. There is also a formula you can use to find the slope of a line, which is usually represented by the letter m. To use this formula, you need the coordinates of two different points on the line. WORDS FORMULA EXAMPLE The slope of a line is the ratio of the difference in y-values to the difference in x-values between any two different points on the line. If (x1 , y1) and (x2 , y2) are any two different points on a line, the slope of the line is m = y2 – y1 x2 – x1 If (2, -3) and (1, 4) are two points on a line, the slope of the line is m = 4 – (-3) = 7 = -7 1 – CONFIDENTIAL

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**Finding Slope by Using the Slope Formula**

1) Find the slope of the line that contains (4, -2) and (-1, 2). m = y2 – y1 x2 – x1 Use the slope formula. = 2 – (-2) -1 – 4 Substitute (4, -2) for ( x1 , y1 ) and (-1, 2) for ( x2 , y2 ) . = 4 -5 Simplify. = -4 5 The slope of the line that contains ( 4, -2) and (-1, 2) is -4. 5 CONFIDENTIAL

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**1a) Find the slope of the line that contains (-2, -2) and (7, -2).**

Now you try! 1a) Find the slope of the line that contains (-2, -2) and (7, -2). 1a) Find the slope of the line that contains (5, -7) and (6, -4). CONFIDENTIAL

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**Sometimes you are not given two points to use in the formula**

Sometimes you are not given two points to use in the formula. You might have to choose two points from a graph or a table. CONFIDENTIAL

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**Finding Slope from Graphs and Tables**

2a) Each graph or table shows a linear relationship. Find the slope. Let (2, 2) for ( x1 , y1 ) and (-2, -1) for ( x2 , y2 ) . m = y2 – y1 x2 – x1 Use the slope formula. = -1 – 2 -2 – 2 Substitute (2, 2) for ( x1 , y1 ) and (-2, -1) for ( x2 , y2 ) . = -3 -4 Simplify. = 3 4 CONFIDENTIAL

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**2b) Each graph or table shows a linear relationship. Find the slope.**

Finding Rates of Change from a Graph 2b) Each graph or table shows a linear relationship. Find the slope. Step1: Choose any two points from the table. Let (2, 0) be (x1 , y1 ) and (2, 3) be (x2 , y2). Step2: Use the slope formula. m = y2 – y1 x2 – x1 Use the slope formula. = 3 – 0 2 – 2 Substitute (2, 0) for ( x1 , y1 ) and (2, 3) for ( x2 , y2 ) . = 3 Simplify. The slope is undefined. CONFIDENTIAL

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**Each graph or table shows a linear relationship. Find the slope.**

Now you try! Each graph or table shows a linear relationship. Find the slope. 2a) 2a) CONFIDENTIAL

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**Remember that slope is a rate of change**

Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how quantity is changing. CONFIDENTIAL

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**Step1: Use the slope formula.**

Application The graph shows how much water is in a reservoir at different times. Find the slope of the line. Then tell what the slope represents. Step1: Use the slope formula. m = y2 – y1 x2 – x1 = 2000 – 3000 60 – 20 = -1000 40 CONFIDENTIAL Next slide

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**In this situation, y represents volume of water and x represents time.**

Step2: Tell what the slope represents. In this situation, y represents volume of water and x represents time. change in volume change in time So slope represents in units of thousands_of cubic_fee_ change in time A slope of -25 means the amount of water in the reservoir is decreasing (negative change) at a rate of 25 thousand cubic feet each hour. CONFIDENTIAL

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Now you try! 3) The graph shows the height of a plant over a period of days. Find the slope of the line. Then tell what the slope represents. CONFIDENTIAL

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If you know the equation that describes a line, you can find its slope by using any two ordered-pair solutions. It is often easiest to use the ordered pairs that contain the intercepts. CONFIDENTIAL

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**Finding Slope from an Equation**

4) Find the slope of the line described by 6x - 5y = 30. Step1: Find the x-intercept. 6x - 5y = 30 6x - 5 (0) = 30 Let y = 0. 6x = 30 6x = 30 x = 5 CONFIDENTIAL

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**Step2: Find the y-intercept.**

6x - 5y = 30 6 (0) - 5y = 30 Let x = 0. -5y = 30 -5y = 30 y = -6 Step1: The line contains (5, 0) and (0, - 6) . Use the slope formula. m = y2 – y1 = - 6 – 0 = -6 = 6 x2 – x – CONFIDENTIAL

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**4) Find the slope of the line described by 2x + 3y = 12.**

Now you try! 4) Find the slope of the line described by 2x + 3y = 12. CONFIDENTIAL

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BREAK CONFIDENTIAL

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Assessment Find the slope of the line that contains each pair of points. 1) (3, 6) and (6, 9) 2) 3, and 1, 2 CONFIDENTIAL

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**Each graph or table shows a linear relationship. Find the slope.**

3) 4) CONFIDENTIAL

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**Find the slope of each line. Then tell what the slope represents.**

5) 6) CONFIDENTIAL

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**Find the slope of the line described by each equation.**

7) 8x + 2y = 96 8) 5x = y CONFIDENTIAL

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**9) The equation 2y + 3x = -6 describes a line with what slope?**

10) A line with slope – 1 could pass through which 3 of the following pairs of points? CONFIDENTIAL

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**The Slope Formula Let’s review**

In the previous lesson, slope was described as the constant rate of change of a line. You saw how to find the slope of a line by using its graph. There is also a formula you can use to find the slope of a line, which is usually represented by the letter m. To use this formula, you need the coordinates of two different points on the line. WORDS FORMULA EXAMPLE The slope of a line is the ratio of the difference in y-values to the difference in x-values between any two different points on the line. If (x1 , y1) and (x2 , y2) are any two different points on a line, the slope of the line is m = y2 – y1 x2 – x1 If (2, -3) and (1, 4) are two points on a line, the slope of the line is m = 4 – (-3) = 7 = -7 1 – CONFIDENTIAL

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**Finding Slope by Using the Slope Formula**

1) Find the slope of the line that contains (4, -2) and (-1, 2). m = y2 – y1 x2 – x1 Use the slope formula. = 2 – (-2) -1 – 4 Substitute (4, -2) for ( x1 , y1 ) and (-1, 2) for ( x2 , y2 ) . = 4 -5 Simplify. = -4 5 The slope of the line that contains ( 4, -2) and (-1, 2) is -4. 5 CONFIDENTIAL

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**Finding Slope from Graphs and Tables**

2a) Each graph or table shows a linear relationship. Find the slope. Let (2, 2) for ( x1 , y1 ) and (-2, -1) for ( x2 , y2 ) . m = y2 – y1 x2 – x1 Use the slope formula. = -1 – 2 -2 – 2 Substitute (2, 2) for ( x1 , y1 ) and (-2, -1) for ( x2 , y2 ) . = -3 -4 Simplify. = 3 4 CONFIDENTIAL

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**2b) Each graph or table shows a linear relationship. Find the slope.**

Finding Rates of Change from a Graph 2b) Each graph or table shows a linear relationship. Find the slope. Step1: Choose any two points from the table. Let (2, 0) be (x1 , y1 ) and (2, 3) be (x2 , y2). Step2: Use the slope formula. m = y2 – y1 x2 – x1 Use the slope formula. = 3 – 0 2 – 2 Substitute (2, 0) for ( x1 , y1 ) and (2, 3) for ( x2 , y2 ) . = 3 Simplify. The slope is undefined. CONFIDENTIAL

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**Step1: Use the slope formula.**

Application The graph shows how much water is in a reservoir at different times. Find the slope of the line. Then tell what the slope represents. Step1: Use the slope formula. m = y2 – y1 x2 – x1 = 2000 – 3000 60 – 20 = -1000 40 CONFIDENTIAL Next slide

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**In this situation, y represents volume of water and x represents time.**

Step2: Tell what the slope represents. In this situation, y represents volume of water and x represents time. change in volume change in time So slope represents in units of thousands_of cubic_fee_ change in time A slope of -25 means the amount of water in the reservoir is decreasing (negative change) at a rate of 25 thousand cubic feet each hour. CONFIDENTIAL

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**Finding Slope from an Equation**

4) Find the slope of the line described by 6x - 5y = 30. Step1: Find the x-intercept. 6x - 5y = 30 6x - 5 (0) = 30 Let y = 0. 6x = 30 6x = 30 x = 5 CONFIDENTIAL

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**Step2: Find the y-intercept.**

6x - 5y = 30 6 (0) - 5y = 30 Let x = 0. -5y = 30 -5y = 30 y = -6 Step1: The line contains (5, 0) and (0, - 6) . Use the slope formula. m = y2 – y1 = - 6 – 0 = -6 = 6 x2 – x – CONFIDENTIAL

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**You did a great job today!**

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