Lesson 19: The Graph of a Linear Equation in Two Variables is a Line
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1 Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Classwork 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 l be the line 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 + k. 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.
2 b. Proof of (2): Let S be any point on line l, as shown. Show that S is a solution to y = mx + b. 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?
3 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?
4 c. When x = 1, what is the value of y? Does this solution appear to be a point on the line? 2 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 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?
6 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.
7 Lesson 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. Example: Graph the equation: 2x + 3y = 9. Replace x with zero and solve for y to determine the y-intercept: The y-intercept is at (0, 3). 2(0) + 3y = 9 3y = 9 y = 3 Replace y with zero and solve for x to determine the x-intercept: The x-intercept is at ( 9 2, 0). 2x + 3(0) = 9 2x = 9 x = 9 2 Problem Set Graph each of the equations in the Problem Set on a different pair of x and y axes. 1. Graph the equation: y = 6x Graph the equation: 9x + 3y = Graph the equation: y = 4x Graph the equation: y = 5 x
8 5. Graph the equation: 3 x + y = Graph the equation: 2x 4y = Graph the equation: y = 3. What is the slope of the graph of this line? 8. Graph the equation: x = 4. What is the slope of the graph of this line? 9. Is the graph of 4x + 5y = 3 a line? Explain Is the graph of 6x 2 2y = 7 a line? Explain.
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