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Chapter 5 The Slope Formula

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The document is a lesson on calculating slope from various representations of lines, including two points, graphs, tables, and equations. It provides examples of finding the slope using different methods and interpreting what the slope represents in real-world contexts. Key methods covered are using the slope formula with two points, choosing points from graphs and tables, and finding the x- and y-intercepts of a line from its equation to then apply the slope formula.

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Lesson Quiz Lesson Presentation Warm Up 5-4 The Slope Formula Holt Algebra 1
Warm Up Add or subtract. 1. 4 + (–6) 2. –3 + 5 3. –7 – 7 4. 2 – (–1) Find the x- and y- intercepts. 5. x + 2 y = 8 6. 3 x + 5 y = –15 x- intercept: –5; y- intercept: –3 x- intercept: 8; y- intercept: 4 – 2 2 3 – 14
Find slope by using the slope formula. Objective
In Lesson 5-3, 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.
 
Example 1: Finding Slope by Using the Slope Formula Find the slope of the line that contains (2, 5) and (8, 1). Use the slope formula. Substitute (2, 5) for (x 1 , y 1 ) and (8, 1) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (2, 5) and (8, 1) is .
Find the slope of the line that contains (–2, –2) and (7, –2). Check It Out! Example 1a Use the slope formula. Substitute (–2, – 2) for (x 1 , y 1 ) and (7, –2) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (–2, –2) and (7, –2) is 0. = 0
Find the slope of the line that contains (5, –7) and (6, –4). Check It Out! Example 1b Use the slope formula. Substitute (5, –7) for (x 1 , y 1 ) and (6, –4) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (5, –7) and (6, –4) is 3. = 3
Find the slope of the line that contains and Check It Out! Example 1c Use the slope formula. Substitute for (x 1 , y 1 ) and for (x 2 , y 2 ) and simplify. The slope of the line that contains and is 2.
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.
Example 2A: Finding Slope from Graphs and Tables The graph shows a linear relationship. Find the slope. Let (0, 2) be (x 1 , y 1 ) and (–2, –2) be (x 2 , y 2 ) . Simplify. Use the slope formula. Substitute (0, 2) for (x 1 , y 1 ) and (–2, –2) for (x 2 , y 2 ) .
Example 2B: Finding Slope from Graphs and Tables The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 1) be ( x 1 , y 1 ) and (–2, 5) be ( x 2 , y 2 ) . Step 2 Use the slope formula. The slope equals −2 Use the slope formula. Simplify. Substitute (0, 1) for and ( – 2, 5) for .
Check It Out! Example 2a The graph shows a linear relationship. Find the slope. Simplify. Use the slope formula. Let (2, 2) be (x 1 , y 1 ) and (4, 3) be (x 2 , y 2 ) . Substitute (2, 2) for (x 1 , y 1 ) and (4, 3) for (x 2 , y 2 ) .
Check It Out! Example 2b Simplify. Use the slope formula. Let (–2, 4) be (x 1 , y 1 ) and (0, –2) be (x 2 , y 2 ) . Substitute (–2, 4) for (x 1 , y 1 ) and (0, –2) for (x 2 , y 2 ) . The graph shows a linear relationship. Find the slope.
Check It Out! Example 2c The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 1) be ( x 1 , y 1 ) and (2, 5) be ( x 2 , y 2 ) . Step 2 Use the slope formula. Use the slope formula. Simplify. Substitute (0, 1) for (x 1 , y 1 ) and (2, 5) for (x 2 , y 2 ) .
Check It Out! Example 2d The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 0) be ( x 1 , y 1 ) and (–2, 3) be ( x 2 , y 2 ) . Step 2 Use the slope formula. Use the slope formula. Simplify Substitute (0, 0) for (x 1 , y 1 ) and (–2, 3) for (x 2 , y 2 ) .
Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how a quantity is changing.
Example 3: Application The graph shows the average electricity costs (in dollars) for operating a refrigerator for several months. Find the slope of the line. Then tell what the slope represents. Step 1 Use the slope formula.
Example 3 Continued Step 2 Tell what the slope represents. In this situation y represents the cost of electricity and x represents time . A slope of 6 mean the cost of running the refrigerator is a rate of 6 dollars per month. So slope represents in units of .
Check It Out! Example 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. Step 1 Use the slope formula.
Check It Out! Example 3 Step 2 Tell what the slope represents. In this situation y represents the height of the plant and x represents time . So slope represents in units of . A slope of mean the plant grows at rate of 1 centimeter every two days.
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.
Example 4: Finding Slope from an Equation Find the slope of the line described by 4 x – 2 y = 16. Step 1 Find the x -intercept. Step 2 Find the y -intercept. 4 x – 2 y = 16 Step 3 The line contains ( 4 , 0 ) and ( 0 , –8 ). Use the slope formula. 4 x – 2 y = 16 4 x = 16 x = 4 – 2 y = 16 y = –8 4 x – 2 (0) = 16 Let y = 0. 4 (0) – 2 y = 16 Let x = 0.
Check It Out! Example 4 Find the slope of the line described by 2 x + 3 y = 12. Step 1 Find the x -intercept. Step 2 Find the y -intercept. 2 x + 3 y = 12 2 x + 3 y = 12 Step 3 The line contains ( 6 , 0 ) and ( 0 , 4 ). Use the slope formula. 2 x + 3 (0) = 12 Let y = 0. 2 (0) + 3 y = 12 Let x = 0. 2 x = 12 x = 6 3 y = 12 y = 4
Lesson Quiz 1. Find the slope of the line that contains (5, 3) and (–1, 4). 2. Find the slope of the line. Then tell what the slope represents. 50; speed of bus is 50 mi/h 3. Find the slope of the line described by x + 2 y = 8.

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Chapter 5 The Slope Formula

  • 1. Lesson Quiz Lesson Presentation Warm Up 5-4 The Slope Formula Holt Algebra 1
  • 2. Warm Up Add or subtract. 1. 4 + (–6) 2. –3 + 5 3. –7 – 7 4. 2 – (–1) Find the x- and y- intercepts. 5. x + 2 y = 8 6. 3 x + 5 y = –15 x- intercept: –5; y- intercept: –3 x- intercept: 8; y- intercept: 4 – 2 2 3 – 14
  • 3. Find slope by using the slope formula. Objective
  • 4. In Lesson 5-3, 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.
  • 5.  
  • 6. Example 1: Finding Slope by Using the Slope Formula Find the slope of the line that contains (2, 5) and (8, 1). Use the slope formula. Substitute (2, 5) for (x 1 , y 1 ) and (8, 1) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (2, 5) and (8, 1) is .
  • 7. Find the slope of the line that contains (–2, –2) and (7, –2). Check It Out! Example 1a Use the slope formula. Substitute (–2, – 2) for (x 1 , y 1 ) and (7, –2) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (–2, –2) and (7, –2) is 0. = 0
  • 8. Find the slope of the line that contains (5, –7) and (6, –4). Check It Out! Example 1b Use the slope formula. Substitute (5, –7) for (x 1 , y 1 ) and (6, –4) for (x 2 , y 2 ) . Simplify. The slope of the line that contains (5, –7) and (6, –4) is 3. = 3
  • 9. Find the slope of the line that contains and Check It Out! Example 1c Use the slope formula. Substitute for (x 1 , y 1 ) and for (x 2 , y 2 ) and simplify. The slope of the line that contains and is 2.
  • 10. 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.
  • 11. Example 2A: Finding Slope from Graphs and Tables The graph shows a linear relationship. Find the slope. Let (0, 2) be (x 1 , y 1 ) and (–2, –2) be (x 2 , y 2 ) . Simplify. Use the slope formula. Substitute (0, 2) for (x 1 , y 1 ) and (–2, –2) for (x 2 , y 2 ) .
  • 12. Example 2B: Finding Slope from Graphs and Tables The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 1) be ( x 1 , y 1 ) and (–2, 5) be ( x 2 , y 2 ) . Step 2 Use the slope formula. The slope equals −2 Use the slope formula. Simplify. Substitute (0, 1) for and ( – 2, 5) for .
  • 13. Check It Out! Example 2a The graph shows a linear relationship. Find the slope. Simplify. Use the slope formula. Let (2, 2) be (x 1 , y 1 ) and (4, 3) be (x 2 , y 2 ) . Substitute (2, 2) for (x 1 , y 1 ) and (4, 3) for (x 2 , y 2 ) .
  • 14. Check It Out! Example 2b Simplify. Use the slope formula. Let (–2, 4) be (x 1 , y 1 ) and (0, –2) be (x 2 , y 2 ) . Substitute (–2, 4) for (x 1 , y 1 ) and (0, –2) for (x 2 , y 2 ) . The graph shows a linear relationship. Find the slope.
  • 15. Check It Out! Example 2c The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 1) be ( x 1 , y 1 ) and (2, 5) be ( x 2 , y 2 ) . Step 2 Use the slope formula. Use the slope formula. Simplify. Substitute (0, 1) for (x 1 , y 1 ) and (2, 5) for (x 2 , y 2 ) .
  • 16. Check It Out! Example 2d The table shows a linear relationship. Find the slope. Step 1 Choose any two points from the table. Let (0, 0) be ( x 1 , y 1 ) and (–2, 3) be ( x 2 , y 2 ) . Step 2 Use the slope formula. Use the slope formula. Simplify Substitute (0, 0) for (x 1 , y 1 ) and (–2, 3) for (x 2 , y 2 ) .
  • 17. Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how a quantity is changing.
  • 18. Example 3: Application The graph shows the average electricity costs (in dollars) for operating a refrigerator for several months. Find the slope of the line. Then tell what the slope represents. Step 1 Use the slope formula.
  • 19. Example 3 Continued Step 2 Tell what the slope represents. In this situation y represents the cost of electricity and x represents time . A slope of 6 mean the cost of running the refrigerator is a rate of 6 dollars per month. So slope represents in units of .
  • 20. Check It Out! Example 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. Step 1 Use the slope formula.
  • 21. Check It Out! Example 3 Step 2 Tell what the slope represents. In this situation y represents the height of the plant and x represents time . So slope represents in units of . A slope of mean the plant grows at rate of 1 centimeter every two days.
  • 22. 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.
  • 23. Example 4: Finding Slope from an Equation Find the slope of the line described by 4 x – 2 y = 16. Step 1 Find the x -intercept. Step 2 Find the y -intercept. 4 x – 2 y = 16 Step 3 The line contains ( 4 , 0 ) and ( 0 , –8 ). Use the slope formula. 4 x – 2 y = 16 4 x = 16 x = 4 – 2 y = 16 y = –8 4 x – 2 (0) = 16 Let y = 0. 4 (0) – 2 y = 16 Let x = 0.
  • 24. Check It Out! Example 4 Find the slope of the line described by 2 x + 3 y = 12. Step 1 Find the x -intercept. Step 2 Find the y -intercept. 2 x + 3 y = 12 2 x + 3 y = 12 Step 3 The line contains ( 6 , 0 ) and ( 0 , 4 ). Use the slope formula. 2 x + 3 (0) = 12 Let y = 0. 2 (0) + 3 y = 12 Let x = 0. 2 x = 12 x = 6 3 y = 12 y = 4
  • 25. Lesson Quiz 1. Find the slope of the line that contains (5, 3) and (–1, 4). 2. Find the slope of the line. Then tell what the slope represents. 50; speed of bus is 50 mi/h 3. Find the slope of the line described by x + 2 y = 8.