Q: A chemist is mixing two solutions, one with 10% salt and one with 20% salt, to obtain 100 mL of a solution with 15% salt. Which system of equations models this scenario?

$\begin{cases} x + y = 100 \\ 0.10x + 0.20y = 0.15 \times 100 \end{cases}$

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 a system of linear equations in two variables?

Accepted Answer

A collection of two or more linear equations in two variables considered simultaneously.

Question 2

Given the system of equations:

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

What is the solution to the system?

Accepted Answer

$(x, y) = (4, 1)$

Question 3

If two lines in a system of equations have the same slope but different $y$-intercepts, what can be said about the system?

Accepted Answer

The system is inconsistent and independent.

Question 4

Which step comes first when using the substitution method to solve a system of equations?

Accepted Answer

Choose an equation and solve for one variable in terms of the other.

Question 5

Consider the system:

$\begin{cases} 3x + 2y = 12 \ 6x + 4y = 24 \end{cases}$

What is true about the solution to this system?

Accepted Answer

There are infinitely many solutions.

Question 6

Which of the following is NOT a step in the elimination method for solving systems of equations?

Accepted Answer

Graph both equations and find the intersection point.

Question 7

A chemist is mixing two solutions, one with 10% salt and one with 20% salt, to obtain 100 mL of a solution with 15% salt. Which system of equations models this scenario?

Accepted Answer

$\begin{cases} x + y = 100 \ 0.10x + 0.20y = 0.15 	imes 100 \end{cases}$

Question 8

Which of the following systems is inconsistent?

Accepted Answer

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

Question 9

A system of equations is said to be dependent if:

Accepted Answer

The equations represent the same line.

Question 10

Which of the following best describes the solution to a system of two linear equations in two variables?

Accepted Answer

The set of all ordered pairs that satisfy both equations.

Question 11

Given the system:

$\begin{cases} x - 2y = 4 \ 2x - 4y = 8 \end{cases}$

What is the nature of its solution?

Accepted Answer

Infinitely many solutions; the system is dependent.

Question 12

Which of the following is a correct first step in solving the system below by elimination?

$\begin{cases} 2x + 3y = 7 \ 4x + 6y = 14 \end{cases}$

Accepted Answer

Subtract the first equation from the second.

Question 13

A medical researcher is tracking two drugs in the bloodstream. Drug A decreases at a rate of $5$ mg/hr, and Drug B at $3$ mg/hr. If the initial amounts are $50$ mg and $30$ mg, respectively, after how many hours will the amounts be equal?

Accepted Answer

$5$ hours

Question 14

Which of the following is NOT a recommended step in the five-step strategy for solving applied problems using systems of equations?

Accepted Answer

Guess the values of the variables before reading the problem.

Question 15

If the graphs of two linear equations intersect at a single point, what does this indicate about the system?

Accepted Answer

The system is consistent and independent.

Question 16

Solve the system using substitution:

$\begin{cases} y = 2x + 1 \ 3x + y = 10 \end{cases}$

Accepted Answer

$(x, y) = (3, 7)$

Question 17

Which of the following systems represents two parallel lines?

Accepted Answer

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

Question 18

A store sells pens for $\$2$ each and notebooks for $\$3$ each. If a customer buys $5$ items for $\$12$, which system of equations models this situation?

Accepted Answer

$\begin{cases} x + y = 5 \ 2x + 3y = 12 \end{cases}$

Question 19

Which of the following is a necessary condition for using the elimination method effectively?

Accepted Answer

The coefficients of one variable must be equal and opposite in sign.

Systems of Linear Equations in Two Variables: Definitions, Methods, and Applications

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