Q: A chemist is mixing two solutions. The first solution contains 40% acid and the second contains 60% acid. She wants to make 100 mL of a solution that is 50% acid. Which system of equations represents this scenario if $$x is $$the amount of the first solution and $$y is $$the amount of the second?

$$\begin{cases} x + y = 100 \\ 0.4x + 0.6y = 50 \end{cases}$$

Q: A business sells two types of tickets: adult tickets for $$\10$ each and child tickets for $$\$$6$$ each. If 50 tickets are sold for a total of $$\400$, which system of equations models this situation?

$$\begin{cases} x + y = 50 \\ 10x + 6y = 400 \end{cases}$$

Q: Which of the following systems is best solved by the elimination method?

$$\begin{cases} 2x + 3y = 7 \\ 4x - 3y = 5 \end{cases}$$

Q: A doctor prescribes two medications. Each pill of Med A contains 200 mg of active ingredient, and each pill of Med B contains 500 mg. If a patient needs 2000 mg total from 6 pills, which system models this scenario?

$$\begin{cases} x + y = 6 \\ 200x + 500y = 2000 \end{cases}$$

Q: A store sells pencils and pens. Pencils cost $$\0.50$ each and pens cost $$\$$1.00$$ each. If a customer buys 10 items for $$\7.00$, which system of equations represents this situation?

$$\begin{cases} x + y = 10 \\ 0.5x + 1y = 7 \end{cases}$$

Question 1

Which of the following best describes the solution to a system of two linear equations when the lines intersect at exactly one point?

Accepted Answer

There is exactly one solution.

Question 2

Given the system of equations:
$$
\begin{cases}
y = 2x + 1 \
y = -x + 4
\end{cases}
$$
What is the solution to the system?

Accepted Answer

$$(1, 3)$$

Question 3

Which of the following systems has no solution?

Accepted Answer

$$y = 2x + 3$$ and $$y = 2x - 4$$

Question 4

A chemist is mixing two solutions. The first solution contains 40% acid and the second contains 60% acid. She wants to make 100 mL of a solution that is 50% acid. Which system of equations represents this scenario if $$x is $$the amount of the first solution and $$y is $$the amount of the second?

Accepted Answer

$$\begin{cases} x + y = 100 \ 0.4x + 0.6y = 50 \end{cases}$$

Question 5

Which method is generally most efficient for solving the system below?
$$
\begin{cases}
y = 3x - 2 \
2x + y = 7
\end{cases}
$$

Accepted Answer

Substitution

Question 6

Given the system:
$$
\begin{cases}
3x + 2y = 12 \
6x + 4y = 24
\end{cases}
$$
How many solutions does this system have?

Accepted Answer

Infinitely many solutions

Question 7

A business sells two types of tickets: adult tickets for $$\10$ each and child tickets for $$\$$6$$ each. If 50 tickets are sold for a total of $$\400$, which system of equations models this situation?

Accepted Answer

$$\begin{cases} x + y = 50 \ 10x + 6y = 400 \end{cases}$$

Question 8

Which of the following systems is best solved by the elimination method?

Accepted Answer

$$\begin{cases} 2x + 3y = 7 \ 4x - 3y = 5 \end{cases}$$

Question 9

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

Accepted Answer

The system has no solution.

Question 10

Solve the following system using elimination:
$$
\begin{cases}
2x + 3y = 13 \
4x + 6y = 26
\end{cases}
$$

Accepted Answer

Infinitely many solutions

Question 11

A doctor prescribes two medications. Each pill of Med A contains 200 mg of active ingredient, and each pill of Med B contains 500 mg. If a patient needs 2000 mg total from 6 pills, which system models this scenario?

Accepted Answer

$$\begin{cases} x + y = 6 \ 200x + 500y = 2000 \end{cases}$$

Question 12

Which of the following is the correct first step to solve the system below by substitution?
$$
\begin{cases}
y = 4x - 5 \
2x + y = 7
\end{cases}
$$

Accepted Answer

Substitute $$y = 4x - 5$$ into $$2x + y = 7$$

Question 13

Given the system:
$$
\begin{cases}
3x + y = 10 \
- x + y = 10
\end{cases}
$$
What is the solution?

Accepted Answer

$$(0, 10)$$

Question 14

Which of the following systems is consistent and dependent?

Accepted Answer

$$y = 2x + 3$$ and $$2y = 4x + 6$$

Question 15

A store sells pencils and pens. Pencils cost $$\0.50$ each and pens cost $$\$$1.00$$ each. If a customer buys 10 items for $$\7.00$, which system of equations represents this situation?

Accepted Answer

$$\begin{cases} x + y = 10 \ 0.5x + 1y = 7 \end{cases}$$

Question 16

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

Accepted Answer

Exactly two solutions

Systems of Linear Equations: Graphing, Substitution, and Elimination

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