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Lesson Plans and Worksheets for Grade 8

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Common Core For Grade 8

### New York State Common Core Math Module 4, Grade 8, Lesson 20

### Lesson 20 Student Outcomes

• Students know that any non-vertical line is the graph of a linear equation in the form of y = mx + b, where b is a constant.

• Students write the equation that represents the graph of a line.

### Lesson 20 Summary

• Write the equation of a line by determining the y-intercept, (0, b) and the slope, m, and replacing the numbers b and m into the equation y = mx + b

Lesson 20 Opening Exercise

Find the equations of the lines for graph 1 and graph 2.

Exercise 1

Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b), where b is a constant.

1. Prove the theorem by completing parts (a)–(c). Given two distinct points, P and Q, on the graph of y = mx + b and let be the line l passing through P and Q. You must show:

(1) any point on the graph of y = mx + b is on line l, and

(2) any point on the line l is on the graph of y = mx + b.

a. Proof of (1): Let R be any point on the graph of y = mx + b. Show that R is on l. Begin by assuming it is not. Assume the graph looks like the diagram below where R is on l'.

What is the slope of line l?

What is the slope of line l'?

What can you conclude about lines l and l'? Explain.

b. Proof of (2): Let S be any point on line l, as shown.

Show that S is a solution to Hint: Use the point (0, b).

c. Now that you have shown that any point on the graph of y = mx + b is on line l (part a), and any point on line l is on the graph of y = mx + b (part b), what can you conclude about the graphs of linear equations?

2. Use x = 4 and x = -4 to find two solutions to the equation x + 2y = 6. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation x + 2y = 6.

b. When x = 1, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -3, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (3, 2) on the line?

e. Is the point (3, 2) a solution to the linear equation x + 2y = 6?

3. Use x = 4 and x = 1 to find two solutions to the equation 3x - y = 9. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation 3x - y = 9.

b. When x = 4.5, what is the value of y? Does this solution appear to be a point on the line?

c. When x = 1/2, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (2, 4) on the line?

e. Is the point (2, 4) a solution to the linear equation 3x - y = 9?

4. Use x = 3 and x = -3 to find two solutions to the equation 2x + 3y = 12. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation 2x + 3y = 12.

b. When x = 2, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -2, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (8, -3) on the line?

e. Is the point (8, -3) a solution to the linear equation 2x + 3y = 12.

5. Use x = 4 and x = -4 to find two solutions to the equation x - 2y = 8. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation x - 2y = 8.

b. When x = 7, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -3, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (-2, -3) on the line?

e. Is the point (-2, -3) a solution to the linear equation x - 2y = 8?

6. Based on your work in Exercises 2–5, what conclusions can you draw about the points on a line and solutions to a linear equation?

7. Based on your work in Exercises 2–5, will a point that is not a solution to a linear equation be a point on the graph of a linear equation? Explain.

8. Based on your work in Exercises 2–5, what conclusions can you draw about the graph of a linear equation?

9. Graph the equation -3x + 8y = 24 using intercepts.

10. Graph the equation x - 6y = 15 using intercepts.

11. Graph the equation 4x + 3y = 21 using intercepts.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Lesson Plans and Worksheets for Grade 8

Lesson Plans and Worksheets for all Grades

More Lessons for Grade 8

Common Core For Grade 8

Videos to help Grade 8 students learn how to prove that any point on the graph of y = mx + b is on a line l and that any point on a line l is a point on the graph of y = mx + b.

• Students know that any non-vertical line is the graph of a linear equation in the form of y = mx + b, where b is a constant.

• Students write the equation that represents the graph of a line.

• Write the equation of a line by determining the y-intercept, (0, b) and the slope, m, and replacing the numbers b and m into the equation y = mx + b

Lesson 20 Opening Exercise

Find the equations of the lines for graph 1 and graph 2.

Exercise 1

Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b), where b is a constant.

1. Prove the theorem by completing parts (a)–(c). Given two distinct points, P and Q, on the graph of y = mx + b and let be the line l passing through P and Q. You must show:

(1) any point on the graph of y = mx + b is on line l, and

(2) any point on the line l is on the graph of y = mx + b.

a. Proof of (1): Let R be any point on the graph of y = mx + b. Show that R is on l. Begin by assuming it is not. Assume the graph looks like the diagram below where R is on l'.

What is the slope of line l?

What is the slope of line l'?

What can you conclude about lines l and l'? Explain.

b. Proof of (2): Let S be any point on line l, as shown.

Show that S is a solution to Hint: Use the point (0, b).

c. Now that you have shown that any point on the graph of y = mx + b is on line l (part a), and any point on line l is on the graph of y = mx + b (part b), what can you conclude about the graphs of linear equations?

2. Use x = 4 and x = -4 to find two solutions to the equation x + 2y = 6. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation x + 2y = 6.

b. When x = 1, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -3, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (3, 2) on the line?

e. Is the point (3, 2) a solution to the linear equation x + 2y = 6?

3. Use x = 4 and x = 1 to find two solutions to the equation 3x - y = 9. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation 3x - y = 9.

b. When x = 4.5, what is the value of y? Does this solution appear to be a point on the line?

c. When x = 1/2, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (2, 4) on the line?

e. Is the point (2, 4) a solution to the linear equation 3x - y = 9?

4. Use x = 3 and x = -3 to find two solutions to the equation 2x + 3y = 12. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation 2x + 3y = 12.

b. When x = 2, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -2, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (8, -3) on the line?

e. Is the point (8, -3) a solution to the linear equation 2x + 3y = 12.

5. Use x = 4 and x = -4 to find two solutions to the equation x - 2y = 8. Plot the solutions as points on the coordinate plane and connect the points to make a line.

a. Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation x - 2y = 8.

b. When x = 7, what is the value of y? Does this solution appear to be a point on the line?

c. When x = -3, what is the value of y? Does this solution appear to be a point on the line?

d. Is the point (-2, -3) on the line?

e. Is the point (-2, -3) a solution to the linear equation x - 2y = 8?

6. Based on your work in Exercises 2–5, what conclusions can you draw about the points on a line and solutions to a linear equation?

7. Based on your work in Exercises 2–5, will a point that is not a solution to a linear equation be a point on the graph of a linear equation? Explain.

8. Based on your work in Exercises 2–5, what conclusions can you draw about the graph of a linear equation?

9. Graph the equation -3x + 8y = 24 using intercepts.

10. Graph the equation x - 6y = 15 using intercepts.

11. Graph the equation 4x + 3y = 21 using intercepts.

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