Q: A chemist is mixing two solutions. The first solution contains $$x$$ liters of 10% acid and the second contains $$y$$ liters of 20% acid. If the total volume is 10 liters and the mixture must be 15% acid, which system represents this scenario?

$$\begin{cases} x + y = 10 \\ 0.10x + 0.20y = 1.5 \end{cases}$$

Q: A business sells two types of tickets: adult tickets for $$x$$ dollars and child tickets for $$y$$ dollars. If 3 adult and 2 child tickets cost $$38$$, and 2 adult and 4 child tickets cost $$44$$, which system models this scenario?

$$\begin{cases} 3x + 2y = 38 \\ 2x + 4y = 44 \end{cases}$$

Q: Which of the following systems is inconsistent?

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

Question 1

Which of the following best describes the main goal of the addition method when solving a system of linear equations?

Accepted Answer

To obtain two equations whose sum eliminates one variable.

Question 2

Given the system: 
$$\begin{cases} 2x + 3y = 7 \ 4x - 3y = 5 \end{cases}$$
What is the solution using the addition method?

Accepted Answer

$$(3, 0)$$

Question 3

If you add the equations $$x + 2y = 5$$ and $$-x + 3y = 4$$, what is the resulting equation?

Accepted Answer

$$5y = 9$$

Question 4

A chemist is mixing two solutions. The first solution contains $$x$$ liters of 10% acid and the second contains $$y$$ liters of 20% acid. If the total volume is 10 liters and the mixture must be 15% acid, which system represents this scenario?

Accepted Answer

$$\begin{cases} x + y = 10 \ 0.10x + 0.20y = 1.5 \end{cases}$$

Question 5

Which of the following indicates that a system of equations has no solution when using the addition method?

Accepted Answer

You obtain a false statement such as $$0 = 1.$$

Question 6

Solve the following system using the addition method:
$$\begin{cases} 3x + 2y = 12 \ 6x + 4y = 24 \end{cases}$$

Accepted Answer

Infinitely many solutions

Question 7

Which step is NOT part of the addition method for solving systems of equations?

Accepted Answer

Divide both equations by the same nonzero number.

Question 8

Given the system $$\begin{cases} 2x - y = 4 \ 4x + y = 8 \end{cases}$$, what is the solution using the addition method?

Accepted Answer

$$(2, 0)$$

Question 9

A business sells two types of tickets: adult tickets for $$x$$ dollars and child tickets for $$y$$ dollars. If 3 adult and 2 child tickets cost $$38$$, and 2 adult and 4 child tickets cost $$44$$, which system models this scenario?

Accepted Answer

$$\begin{cases} 3x + 2y = 38 \ 2x + 4y = 44 \end{cases}$$

Question 10

If the addition method leads to the equation $$0 = 0$$, what does this indicate about the system?

Accepted Answer

The system has infinitely many solutions.

Question 11

Which of the following systems is inconsistent?

Accepted Answer

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

Question 12

Solve the following system using the addition method:
$$\begin{cases} x + 2y = 8 \ 2x - 2y = 2 \end{cases}$$

Accepted Answer

$$(4, 2)$$

Question 13

Which of the following is the most efficient first step when using the addition method for the system $$\begin{cases} 2x + 3y = 7 \ 4x + 6y = 14 \end{cases}$$?

Accepted Answer

Notice that the second equation is a multiple of the first and conclude infinitely many solutions.

Question 14

A store sells pencils and pens. A customer buys 5 pencils and 3 pens for $$19. Another customer buys 2 pencils and 4 pens for 14. $$What is the price of one pencil and one pen?

Accepted Answer

$$2$$ for a pencil, $$4$$ for a pen

Question 15

Which of the following is NOT a possible outcome when solving a system of linear equations using the addition method?

Accepted Answer

A solution with three variables

Solving Systems of Linear Equations by the Addition Method

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