Q: A chemist is mixing two solutions. The first solution contains $$x$$ grams of salt per liter, and the second contains $$y$$ grams per liter. If she mixes 3 liters of the first with 2 liters of the second to get a mixture with 18 grams of salt per liter, and 2 liters of the first with 4 liters of the second to get a mixture with 20 grams per liter, which system represents this scenario?

$$\begin{cases} 3x + 2y = 18 \\ 2x + 4y = 20 \end{cases}$$

Q: A store sells pens and pencils. Pens cost $$x$$ dollars each and pencils cost $$y$$ dollars each. If 5 pens and 3 pencils cost $$13$$, and 2 pens and 4 pencils cost $$8$$, which system of equations models this situation?

$$\begin{cases} 5x + 3y = 13 \\ 2x + 4y = 8 \end{cases}$$

Q: A car rental company charges a flat fee of $$x$$ dollars and $$y$$ dollars per mile. If a 100-mile trip costs $$150$$ and a 200-mile trip costs $$250$$, which system models this situation?

$$\begin{cases} x + 100y = 150 \\ x + 200y = 250 \end{cases}$$

Q: A business has two phone plans. Plan A charges $$x$$ dollars per month plus $$y$$ dollars per minute. Plan B charges $$2x$$ dollars per month plus $$0.5y$$ dollars per minute. If a customer pays $$50$$ for 20 minutes on Plan A and $$70$$ for 40 minutes on Plan B, which system represents this scenario?

$$\begin{cases} x + 20y = 50 \\ 2x + 0.5 \times 40 = 70 \end{cases}$$

Question 1

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

Accepted Answer

An ordered pair $$(x, y)$$ that satisfies both equations.

Question 2

Given the system of equations:

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

What is the nature of the solution set?

Accepted Answer

There are infinitely many solutions.

Question 3

Which method would be most efficient to solve the following system?

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

Accepted Answer

Addition (Elimination) Method

Question 4

A chemist is mixing two solutions. The first solution contains $$x$$ grams of salt per liter, and the second contains $$y$$ grams per liter. If she mixes 3 liters of the first with 2 liters of the second to get a mixture with 18 grams of salt per liter, and 2 liters of the first with 4 liters of the second to get a mixture with 20 grams per liter, which system represents this scenario?

Accepted Answer

$$\begin{cases} 3x + 2y = 18 \ 2x + 4y = 20 \end{cases}$$

Question 5

If the system $$\begin{cases} x + 2y = 5 \ 2x + 4y = 10 \end{cases} is $$solved by the elimination method, what is the result?

Accepted Answer

Infinitely many solutions

Question 6

A store sells pens and pencils. Pens cost $$x$$ dollars each and pencils cost $$y$$ dollars each. If 5 pens and 3 pencils cost $$13$$, and 2 pens and 4 pencils cost $$8$$, which system of equations models this situation?

Accepted Answer

$$\begin{cases} 5x + 3y = 13 \ 2x + 4y = 8 \end{cases}$$

Question 7

Which of the following is NOT a step in the Substitution Method for solving systems of equations?

Accepted Answer

Add the two equations together to eliminate a variable.

Question 8

Given the system $$\begin{cases} 2x - y = 1 \ 3x + y = 9 \end{cases}$$, what is the solution?

Accepted Answer

$$(2, 3)$$

Question 9

If two lines represented by a system of equations are parallel, what can be said about the solution to the system?

Accepted Answer

There is no solution.

Question 10

A medical researcher is tracking the spread of a disease. In one city, the number of infected people is modeled by $$x + y = 1000$$, and in another city by $$2x - y = 500. $$What is the value of $$x$$ and $$y$$?

Accepted Answer

$$x = 500, y = 500$$

Question 11

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

Accepted Answer

The point where the two lines intersect.

Question 12

When using the elimination method, what is the main goal of multiplying one or both equations by appropriate numbers?

Accepted Answer

To make the coefficients of one variable opposites.

Question 13

A car rental company charges a flat fee of $$x$$ dollars and $$y$$ dollars per mile. If a 100-mile trip costs $$150$$ and a 200-mile trip costs $$250$$, which system models this situation?

Accepted Answer

$$\begin{cases} x + 100y = 150 \ x + 200y = 250 \end{cases}$$

Question 14

If you solve the system $$\begin{cases} x + y = 4 \ x - y = 2 \end{cases} by $$the elimination method, what is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 15

Which of the following is a necessary condition for a system of two linear equations to have exactly one solution?

Accepted Answer

The lines intersect at exactly one point.

Question 16

A business has two phone plans. Plan A charges $$x$$ dollars per month plus $$y$$ dollars per minute. Plan B charges $$2x$$ dollars per month plus $$0.5y$$ dollars per minute. If a customer pays $$50$$ for 20 minutes on Plan A and $$70$$ for 40 minutes on Plan B, which system represents this scenario?

Accepted Answer

$$\begin{cases} x + 20y = 50 \ 2x + 0.5 	imes 40 = 70 \end{cases}$$

Question 17

If you use the substitution method to solve $$\begin{cases} x = 2y + 1 \ 3x - y = 7 \end{cases}$$, what is the value of $$y$$?

Accepted Answer

$$y = 2$$

Question 18

Which of the following real-world scenarios can be modeled by a system of two linear equations in two variables?

Accepted Answer

Finding the intersection point of two roads on a map.

Question 19

A company produces two products, A and B. Each unit of A requires 2 hours of labor and 3 units of material. Each unit of B requires 1 hour of labor and 2 units of material. If the company has 100 hours of labor and 150 units of material available, which system models the constraints?

Accepted Answer

$$\begin{cases} 2x + 1y = 100 \ 3x + 2y = 150 \end{cases}$$

Solving Systems of Linear Equations in Two Variables

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