Q: Which of the following systems is equivalent to the system below? $\begin{cases} 3x - 9y = 15 \\ 5x - 15y = 25 \end{cases}$

$\begin{cases} x - 3y = 5 \\ x - 3y = 5 \end{cases}$

Question 1

Solve the following system of equations using the addition (elimination) method:

$\begin{cases}
3x - 9y = 15 \
5x - 15y = 25
\end{cases}$

Which of the following best describes the solution to this system?

Accepted Answer

There are infinitely many solutions.

Question 2

Given the system of equations:

$\begin{cases}
x - 9y = 15 \
x + 15y = 25
\end{cases}$

What is the solution to this system?

Accepted Answer

The solution is $\left( 20, \frac{5}{3} \right)$.

Question 3

A chemist is mixing two solutions. The first equation represents the amount of chemical A and the second equation represents the amount of chemical B:

$\begin{cases}
3x - 9y = 15 \
5x - 15y = 25
\end{cases}$

What does the solution to this system tell you about the relationship between the two chemicals?

Accepted Answer

The amounts of chemicals are dependent, so there are infinitely many combinations.

Question 4

Which of the following systems is equivalent to the system below?

$\begin{cases}
3x - 9y = 15 \
5x - 15y = 25
\end{cases}$

Accepted Answer

$\begin{cases} x - 3y = 5 \ x - 3y = 5 \end{cases}$

Question 5

If you multiply the first equation $3x - 9y = 15$ by $5$ and the second equation $5x - 15y = 25$ by $3$, what do you notice about the resulting equations?

Accepted Answer

They are equivalent equations.

Question 6

Which of the following best describes a system of equations that has infinitely many solutions?

Accepted Answer

The equations represent the same line.

Question 7

Suppose you are solving a system of equations and, after elimination, you obtain $0 = 0$. What does this indicate about the system?

Accepted Answer

There are infinitely many solutions.

Question 8

A business is trying to solve the following system to determine the number of products to produce:

$\begin{cases}
3x - 9y = 15 \
5x - 15y = 25
\end{cases}$

What should the business conclude?

Accepted Answer

There are infinitely many combinations of products that satisfy both equations.

Question 9

Which of the following is a correct step in using the addition method to solve $\begin{cases} 3x - 9y = 15 \ 5x - 15y = 25 \end{cases}$?

Accepted Answer

Multiply the first equation by $5$ and the second by $3$ to align the $x$ coefficients.

Question 10

If a system of equations has infinitely many solutions, which of the following must be true?

Accepted Answer

The equations are multiples of each other.

Solving a System of Equations Using the Addition (Elimination) Method

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