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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