Q: Which of the following systems of equations has no solution?

$\begin{cases} 2x + 3y = 6 \\ 4x + 6y = 10 \end{cases}$

Q: A standardized test has 125 questions worth a total of 1300 points. True/False questions are worth 8 points each, and Multiple Choice questions are worth 14 points each. Let $T$ be the number of True/False questions and $M$ be the number of Multiple Choice questions. Which system of equations models this scenario?

$\begin{cases} T + M = 125 \\ 8T + 14M = 1300 \end{cases}$

Q: If two lines have the same slope but different $y$-intercepts, what can you conclude about the system of equations they form?

The system has no solution because the lines are parallel.

Q: If a system of equations has exactly one solution, what does this mean about the graphs of the equations?

The lines intersect at one point.

Question 1

A car rental company charges $40 per day plus $0.10 per mile driven. Write a function $C(x)$ that models the total cost of renting a car for $x$ miles, and determine the cost for driving 150 miles in one day.

Accepted Answer

$C(x) = 40 + 0.10x;\ C(150) = 55$

Question 2

Given the piecewise function 
$
  f(x) = \begin{cases} 2x+4 & x < 2 \ -x+2 & x \geq 2 \end{cases}
$
What is the value of $f(3)$?

Accepted Answer

$f(3) = -1$

Question 3

Which of the following systems of equations has no solution?

Accepted Answer

$\begin{cases} 2x + 3y = 6 \ 4x + 6y = 10 \end{cases}$

Question 4

A standardized test has 125 questions worth a total of 1300 points. True/False questions are worth 8 points each, and Multiple Choice questions are worth 14 points each. Let $T$ be the number of True/False questions and $M$ be the number of Multiple Choice questions. Which system of equations models this scenario?

Accepted Answer

$\begin{cases} T + M = 125 \ 8T + 14M = 1300 \end{cases}$

Question 5

If two lines have the same slope but different $y$-intercepts, what can you conclude about the system of equations they form?

Accepted Answer

The system has no solution because the lines are parallel.

Question 6

Solve the following system using substitution:
$
\begin{cases}
x + 2y = 7 \
3x - y = 5
\end{cases}
$

Accepted Answer

$(x, y) = (3, 2)$

Question 7

A rectangular metal sheet is 8 feet longer than it is wide. If the perimeter is 228 feet, what is the width?

Accepted Answer

$w = 53$ feet

Question 8

Which method is generally most efficient for solving a system of equations when both equations are already in the form $Ax + By = C$?

Accepted Answer

Addition/Elimination

Question 9

Given the system:
$
\begin{cases}
4x + 2y = 6 \
2x + y = 3
\end{cases}
$
What is the solution?

Accepted Answer

$(x, y) = (1, 1)$

Question 10

A business has a cost function $C(x) = 0.85x + 350$ and a revenue function $R(x) = 1.1x$. Find the break-even point.

Accepted Answer

$x = 1400$

Question 11

Which of the following best describes the solution to a system of equations whose graphs are the same line?

Accepted Answer

Infinitely many solutions

Question 12

Find the domain of the piecewise function:
$
  f(x) = \begin{cases} 2x+4 & x < 2 \ -x+2 & x \geq 2 \end{cases}
$

Accepted Answer

$(-\infty, \infty)$

Question 13

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

Accepted Answer

The lines intersect at one point.

Question 14

Solve the following system using elimination:
$
\begin{cases}
3x + 2y = 12 \
2x - y = 4
\end{cases}
$

Accepted Answer

$(x, y) = (2, 3)$

Question 15

You are comparing two gym memberships. Gym A charges $25/month plus $5 per visit. Gym B charges $40/month plus $2 per visit. For how many visits per month will the total cost be the same?

Accepted Answer

$5$ visits

B5

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