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Every Line is a Graph of a Linear Equation

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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.

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.



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