Question 1

Given the system of equations:

$$\begin{cases} 2x - 4 = 3 \ 6x - 3y = 9 \end{cases}$$

What is the solution set?

Accepted Answer

There are infinitely many solutions; the solution set is $$\{(x, y) \mid 2x - y = 3\}.$$

Question 2

Which of the following statements is TRUE about a system of two linear equations that results in an identity such as $$0 = 0$$ after elimination?

Accepted Answer

The system has infinitely many solutions.

Question 3

Consider the system:

$$\begin{cases} 2x + 3y = 3 \ 4x + 6y = -6 \end{cases}$$

What can you conclude about the solution?

Accepted Answer

The system has no solution because the lines are parallel.

Question 4

If a system of equations reduces to a false statement such as $$0 = 5$$ after elimination, what does this indicate?

Accepted Answer

The system has no solution.

Question 5

Given the system:

$$\begin{cases} y = 2x - 3 \ y = 2x - 3 \end{cases}$$

What is the solution set?

Accepted Answer

There are infinitely many solutions; all points on $$y = 2x - 3.$$

Question 6

Which of the following best describes a system of equations whose graphs are parallel lines?

Accepted Answer

The system is inconsistent and has no solution.

Question 7

A chemist is mixing two solutions. The first solution is described by $$y = \frac{2}{3}x - 1$$ and the second by $$y = \frac{2}{3}x + 1. $$What does the system of these two equations represent?

Accepted Answer

The solutions are parallel and there is no intersection.

Question 8

Given the system:

$$\begin{cases} y = -\frac{1}{5}x - \frac{4}{5} \ y = -\frac{1}{5}x - \frac{4}{5} \end{cases}$$

What is the nature of the solution?

Accepted Answer

Infinitely many solutions.

Question 9

If both variables are eliminated and the resulting statement is true, what can you conclude about the system?

Accepted Answer

The system has infinitely many solutions.

Question 10

Suppose you are solving a system of equations and after elimination you get $$0 = 0. $$Which of the following is correct?

Accepted Answer

The system is dependent and has infinitely many solutions.

Question 11

Which of the following is a correct interpretation of the solution set $$\{(x, y) \mid 2x - y = 3\}$$?

Accepted Answer

All points $$(x, y)$$ that satisfy $$2x - y = 3.$$

Question 12

A system of equations is called 'dependent' if:

Accepted Answer

It has infinitely many solutions.

Question 13

Which of the following systems is inconsistent?

Accepted Answer

$$\begin{cases} y = 2x + 1 \ y = 2x - 3 \end{cases}$$

Question 14

If a system of equations has exactly one solution, what does this mean about the lines represented by the equations?

Accepted Answer

The lines intersect at exactly one point.

Question 15

Which of the following best describes the solution to the system $$\begin{cases} y = 3x + 2 \ y = 3x - 5 \end{cases}$$?

Accepted Answer

No solution; the lines are parallel.

Question 16

A real-world scenario: Two companies offer cell phone plans. Company A charges $$y = 0.10x + 20$$ per month, and Company B charges $$y = 0.10x + 25. $$What does the system of equations tell you about the plans?

Accepted Answer

The plans never cost the same; the lines are parallel.

Question 17

If you graph two equations and they overlap completely, what does this mean about their solution set?

Accepted Answer

There are infinitely many solutions.

Question 18

Which of the following is NOT a possible outcome when solving a system of two linear equations?

Accepted Answer

Exactly two solutions.

Solving Systems of Linear Equations: Special Cases/Dependent 3.1

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