Q: A chemist is mixing two solutions. The first solution contains $$x$$ grams of chemical A and $$y$$ grams of chemical B. The second solution contains $$2x$$ grams of chemical A and $$y-1$$ grams of chemical B. If the total amount of chemical A is 8 grams and the total amount of chemical B is 5 grams, which system of equations represents this scenario?

$$\begin{cases} x + 2x = 8 \\ y + (y-1) = 5 \end{cases}$$

Q: A business sells two types of tickets: adult tickets for $$x$$ dollars and child tickets for $$y$$ dollars. If 3 adult tickets and 2 child tickets cost $$22, and 2 adult tickets and 4 child tickets cost 20$$, which system of equations models this situation?

$$\begin{cases} 3x + 2y = 22 \\ 2x + 4y = 20 \end{cases}$$

Q: A medical researcher is testing two drugs. The total dosage for a patient is 100 mg, with drug A being twice the amount of drug B. Which system of equations represents this scenario if $$x is $$the amount of drug A and $$y is $$the amount of drug B?

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

Q: Which of the following systems is inconsistent?

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

Question 1

Which of the following ordered pairs is a solution to the system of equations $$\begin{cases} 2x + y = 7 \ x - y = 3 \end{cases}$$?

Accepted Answer

$$(2, 3)$$

Question 2

A chemist is mixing two solutions. The first solution contains $$x$$ grams of chemical A and $$y$$ grams of chemical B. The second solution contains $$2x$$ grams of chemical A and $$y-1$$ grams of chemical B. If the total amount of chemical A is 8 grams and the total amount of chemical B is 5 grams, which system of equations represents this scenario?

Accepted Answer

$$\begin{cases} x + 2x = 8 \ y + (y-1) = 5 \end{cases}$$

Question 3

Which of the following best describes a system of equations whose graphs are parallel lines?

Accepted Answer

No solution; the system is inconsistent.

Question 4

Solve the system using the substitution method: $$\begin{cases} x = y + 2 \ 2x + y = 7 \end{cases}.$$

Accepted Answer

$$(3, 1)$$

Question 5

A business sells two types of tickets: adult tickets for $$x$$ dollars and child tickets for $$y$$ dollars. If 3 adult tickets and 2 child tickets cost $$22, and 2 adult tickets and 4 child tickets cost 20$$, which system of equations models this situation?

Accepted Answer

$$\begin{cases} 3x + 2y = 22 \ 2x + 4y = 20 \end{cases}$$

Question 6

What is the solution to the system $$\begin{cases} x + y = 5 \ x - y = 1 \end{cases}$$ using the elimination method?

Accepted Answer

$$(3, 2)$$

Question 7

If a system of equations has an infinite number of solutions, which of the following must be true?

Accepted Answer

The equations represent the same line.

Question 8

Given the system $$\begin{cases} 2x + 3y = 12 \ 4x + 6y = 24 \end{cases}$$, what is the nature of its solutions?

Accepted Answer

Infinite number of solutions

Question 9

A medical researcher is testing two drugs. The total dosage for a patient is 100 mg, with drug A being twice the amount of drug B. Which system of equations represents this scenario if $$x is $$the amount of drug A and $$y is $$the amount of drug B?

Accepted Answer

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

Question 10

Solve the system by elimination: $$\begin{cases} 3x + 2y = 12 \ 2x - 2y = 4 \end{cases}.$$

Accepted Answer

$$(2, 3)$$

Question 11

Which of the following statements is true about the solution to a system of equations whose graphs coincide?

Accepted Answer

There are infinitely many solutions.

Question 12

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

$$3$$ for a pencil and $$2$$ for a pen

Question 13

Which of the following systems is inconsistent?

Accepted Answer

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

Question 14

Solve the system: $$\begin{cases} x + 2y = 7 \ 2x + 4y = 14 \end{cases}.$$

Accepted Answer

Infinite number of solutions

Question 15

If the solution to a system of equations is $$(4, 2)$$, which of the following must be true?

Accepted Answer

Both equations are satisfied when $$x = 4$$ and $$y = 2.$$

Question 16

A system of equations is given by $$\begin{cases} x + y = 10 \ x - y = 4 \end{cases}. $$What is the value of $$x$$?

Accepted Answer

$$7$$

Question 17

Which of the following is the first step in solving a system by the substitution method?

Accepted Answer

Solve one of the equations for one of its variables.

Question 18

A system of equations is solved and the result is $$0 = -1. $$What does this mean?

Accepted Answer

The system has no solution.

Question 19

Which of the following best describes a consistent and independent system?

Accepted Answer

The system has exactly one solution.

Systems of Linear Equations in Two Variables – Intermediate Algebra Study Notes

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