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Geometric Interpretation of the Solutions of a Linear System

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Examples, solutions, worksheets, and videos to help Grade 8 students learn how to graph two equations and find the point of intersection.

New York State Common Core Math Grade 8, Module 4, Lesson 25

Download Worksheets for Grade 8, Module 4, Lesson 25

Lesson 25 Student Outcomes

• Students graph two equations and find the point of intersection.
• Students identify the point of intersection of the two lines as the solution to the system.
• Students verify by computation that the point of intersection is a solution to each of the equations in the system.

Lesson 25 Student Summary

When a system of linear equations is graphed, the point of intersection of the lines of the graph represents the solution to the system. Two distinct lines intersect at most at one point. The coordinates of that point (x, y) represent values that make both equations of the system true.

Lesson 25 Opening Exercise

Exercises
1. Graph the linear system on a coordinate plane:
2y + x = 12
y = 5/6 x - 25
a. Name the ordered pair where the graphs of the two linear equations intersect.
b. Verify that the ordered pair named in (a) is a solution to 2y + x = 12
c. Verify that the ordered pair named in (a) is a solution to y = 5/6 x - 2
d. Could the point (4, 4) be a solution to the system of linear equations? That is, would (4. 4) make both equations true? Why or why not?

2. Graph the linear system on a coordinate plane:
x + y = -2
y = 4x + 3
a. Name the ordered pair where the graphs of the two linear equations intersect.
b. Verify that the ordered pair named in (a) is a solution to x + y = -2
c. Verify that the ordered pair named in (a) is a solution to y = 4x + 3
d. Could the point (-4, 2) be a solution to the system of linear equations? That is, would (-4, 2) make both equations true? Why or why not?

3. Graph the linear system on a coordinate plane:
3x + y = -3
-2x + y = 2
a. Name the ordered pair where the graphs of the two linear equations intersect.
b. Verify that the ordered pair named in (a) is a solution to 3x + y = -3
c. Verify that the ordered pair named in (a) is a solution to -2x + y = 2
d. Could the point (1, 4) be a solution to the system of linear equations? That is, would (1, 4) make both equations true? Why or why not?




4. Graph the linear system on a coordinate plane:
2x - 3y = 18
2x + y = 2
a. Name the ordered pair where the graphs of the two linear equations intersect.
b. Verify that the ordered pair named in (a) is a solution to 2x - 3y = 18
c. Verify that the ordered pair named in (a) is a solution to 2x + y = 2
d. Could the point (3, -1) be a solution to the system of linear equations? That is, would (3, -1) make both equations true? Why or why not?

5. Graph the linear system on a coordinate plane:
y - x = 3
y = -4x - 2
a. Name the ordered pair where the graphs of the two linear equations intersect.
b. Verify that the ordered pair named in (a) is a solution to y - x = 3
c. Verify that the ordered pair named in (a) is a solution to y = -4x - 2
d. Could the point (-2, 6) be a solution to the system of linear equations? That is, would (-2, 6) make both equations true? Why or why not?

6. Write two different systems of equations with (1, -2) as the solution.



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