Q: Which of the following systems is dependent?

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

Question 1

Which of the following best describes a system of linear equations that has exactly one solution?

Accepted Answer

The lines intersect at exactly one point.

Question 2

Given the system of equations:

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

Is the point $(-4, 16)$ a solution to the system?

Accepted Answer

Yes, because it satisfies both equations.

Question 3

A system of equations is given by:

$\begin{cases} 3x - 2y = -4 \ 9x + 6y = -6 \end{cases}$

What is the nature of the solution to this system?

Accepted Answer

There are no solutions; the system is inconsistent.

Question 4

Which method is generally most efficient for solving a system where one variable has a coefficient of 1 or -1?

Accepted Answer

Substitution method

Question 5

Solve the following system using the addition method:

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

Accepted Answer

$x = 1, y = 3$

Question 6

A chemist is mixing two solutions: one with 12% salt and another with 5% salt. If $x$ liters of the 12% solution and $y$ liters of the 5% solution are mixed to obtain $-0.02$ liters of salt, which system models this situation?

Accepted Answer

$0.12x + 0.05y = -0.02$

Question 7

Given the system:

$\begin{cases} 2x - y = 13 \ -4x - 9y = 7 \end{cases}$

What is the solution?

Accepted Answer

$(5, -3)$

Question 8

Which of the following systems is dependent?

Accepted Answer

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

Question 9

A system of equations is solved and the result is $0 = 0$. What does this indicate about the system?

Accepted Answer

The system has infinitely many solutions.

Question 10

If two lines represented by a system of equations are parallel, what can you conclude about the system?

Accepted Answer

It has no solution.

Question 11

Solve the following system by substitution:

$\begin{cases} x + 2y = -6 \ 2x + y = -3 \end{cases}$

Accepted Answer

$x = -4, y = -1$

Question 12

Which of the following is NOT a possible outcome when solving a system of two linear equations in two variables?

Accepted Answer

Exactly two solutions

Question 13

A medical researcher is tracking two medications in the bloodstream. The first medication decreases at a rate of $0.08x$ per hour, and the second at $0.03y$ per hour. If after one hour, the total decrease is $-0.14$ units, which equation models this scenario?

Accepted Answer

$0.08x - 0.03y = -0.14$

Question 14

Given the system:

$\begin{cases} 5x - 3y = 14 \ 2x - y = 6 \end{cases}$

What is the solution?

Accepted Answer

$(4, 2)$

Question 15

Which of the following best describes an inconsistent system?

Accepted Answer

A system with no solution.

Question 16

A system of equations is solved and the result is $21 = 12$. What does this indicate?

Accepted Answer

The system has no solution.

Question 17

Which method is most appropriate for solving a system with decimal coefficients?

Accepted Answer

Addition (elimination) method

Question 18

Solve the following system:

$\begin{cases} 0.12x + 0.05y = -0.02 \ 0.08x - 0.03y = -0.14 \end{cases}$

What is the solution?

Accepted Answer

$x = -1, y = 2$

Question 19

If a system of equations is dependent, what can be said about the graphs of the equations?

Accepted Answer

The lines coincide (are the same line).

Question 20

A real-world scenario: Two taxi companies charge different rates. Company A charges a flat fee of $\$3$ plus $\$2$ per mile, and Company B charges a flat fee of $\$5$ plus $\$1.50$ per mile. For what number of miles will the cost be the same?

Accepted Answer

$4$ miles

Systems of Linear Equations in Two Variables (Chapter 4, Section 4.1)

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