# Nature of Solutions of a System of Linear Equations

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Lesson Plans and Worksheets for Grade 8
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Examples, videos, worksheets, and solutions to help Grade 8 students learn a strategy for solving a system of linear equations algebraically.

### New York State Common Core Math Grade 8, Module 4, Lesson 27

Lesson 27 Student Outcomes

• Students know that since two equations in the form ax + by = c and a'x + b'y = c' graph as the same line when a'/a = b'/b = c'/c, then the system of linear equations has infinitely many solutions.
• Students know a strategy for solving a system of linear equations algebraically.

Lesson 27 Student Summary

A system of linear equations can have a unique solution, no solutions, or infinitely many solutions. Systems with a unique solution will be comprised of linear equations that have different slopes that graph as distinct lines, intersecting at only one point.

Systems with no solution will be comprised of linear equations that have the same slope that graph as parallel lines (no intersection).

Systems with infinitely many solutions will be comprised of linear equations that have the same slope and - intercept that graph as the same line. When equations graph as the same line, every solution to one equation will also be a solution to the other equation.

A system of linear equations can be solved using a substitution method. That is, if two expressions are equal to the same value, then they can be written equal to one another.

Lesson 27 Opening Exercise

Exercises 1–3
Determine the nature of the solution to each system of linear equations.
1. 3x + 4y = 5
y = -3/4 x + 1

2. 7x + 2y = -4
x - y = 5

3. 9x + 6y = 3
3x + 2y = 1

Example 1
In this example, students realize that graphing a system of equations will yield a solution, but the precise coordinates of the solution cannot be determined from the graph.

Example 2
Does the system
y = 7x - 2
2y - 4x = 10
have a solution?

Example 3
Does the system
4y = 26x + 4
y = 11x - 1
have a solution?

Exercises 4–7
Determine the nature of the solution to each system of linear equations. If the system has a solution, find it algebraically; then, verify that your solution is correct by graphing.
4. 3x + 3y = -21
x + y = -7

5. y = 3/2 x - 1
3y = x + 2

6. x = 12y - 4
x = 9y + 7

7. Write a system of equations with (4, -5) as its solution.

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