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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.

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

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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

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.

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