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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 19

### Lesson 19 Student Outcomes

• Students 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 graph linear equations on the coordinate plane.

### Lesson 19 Summary

• The graph of a linear equation is a line. A linear equation can be graphed using two-points: the x-intercept and the
y-intercept.

Lesson 19 Opening Exercises

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.

Lesson Plans and Worksheets for Grade 8

Lesson Plans and Worksheets for all Grades

More Lessons for Grade 8

Common Core For Grade 8

Examples, solutions, and 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 and how to graph linear equations on the coordinate plane.

• Students graph linear equations on the coordinate plane.

Lesson 19 Opening Exercises

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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