Q: Which of the following systems is dependent?

$$\begin{cases} 3y - 2x = 6 \\ -12y + 8x = -24 \end{cases}$$

Q: Translate the following scenario into a system of equations: The perimeter of a rectangle is 22 in. The width is one less than twice the length. Let $$l be $$the length and $$w$$ the width.

$$\begin{cases} 2l + 2w = 22 \\ w = 2l - 1 \end{cases}$$

Q: A store sells two types of shirts. One costs $$\8.25$ and the other $$\$$11.50. If 52 $$shirts are sold for a total of $$\464.75$, which system of equations represents this situation? Let x$ be the number of $$\$$8.25$$ shirts and $$y$$ the number of $$\11.50$ shirts.

$$\begin{cases} x + y = 52 \\ 8.25x + 11.50y = 464.75 \end{cases}$$

Q: Which of the following systems is consistent and independent?

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

Q: Translate the following scenario into a system of equations: Complementary angles are such that 3 times the smaller angle minus 2 times the larger angle is 20. Let $$x be $$the smaller angle and $$y$$ the larger angle.

$$\begin{cases} x + y = 90 \\ 3x - 2y = 20 \end{cases}$$

Question 1

Which of the following ordered pairs is a solution to the system of equations $$\begin{cases} x + y = 5 \ y = \frac{1}{2}x - 1 \end{cases}$$?

Accepted Answer

$$(4, 1)$$

Question 2

Given the system $$\begin{cases} x - y = 1 \ x + y = 5 \end{cases}$$, what is the solution?

Accepted Answer

$$(3, 2)$$

Question 3

A system of equations is called inconsistent if:

Accepted Answer

It has no solution.

Question 4

Which of the following systems is dependent?

Accepted Answer

$$\begin{cases} 3y - 2x = 6 \ -12y + 8x = -24 \end{cases}$$

Question 5

If two lines are parallel when graphed, what can you conclude about the system of equations they represent?

Accepted Answer

The system is inconsistent and has no solution.

Question 6

Translate the following scenario into a system of equations: The perimeter of a rectangle is 22 in. The width is one less than twice the length. Let $$l be $$the length and $$w$$ the width.

Accepted Answer

$$\begin{cases} 2l + 2w = 22 \ w = 2l - 1 \end{cases}$$

Question 7

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

Accepted Answer

The graphs of the equations intersect at one point.

Question 8

Given the system $$\begin{cases} y = \frac{2}{3}x - 2 \ y = 3x + 5 \end{cases}$$, what is the solution?

Accepted Answer

$$(-3, -4)$$

Question 9

A store sells two types of shirts. One costs $$\8.25$ and the other $$\$$11.50. If 52 $$shirts are sold for a total of $$\464.75$, which system of equations represents this situation? Let x$ be the number of $$\$$8.25$$ shirts and $$y$$ the number of $$\11.50$ shirts.

Accepted Answer

$$\begin{cases} x + y = 52 \ 8.25x + 11.50y = 464.75 \end{cases}$$

Question 10

Which of the following systems is consistent and independent?

Accepted Answer

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

Question 11

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

Accepted Answer

The system is inconsistent and has no solution.

Question 12

Translate the following scenario into a system of equations: Complementary angles are such that 3 times the smaller angle minus 2 times the larger angle is 20. Let $$x be $$the smaller angle and $$y$$ the larger angle.

Accepted Answer

$$\begin{cases} x + y = 90 \ 3x - 2y = 20 \end{cases}$$

Question 13

Which of the following statements is true about dependent equations in a system?

Accepted Answer

One equation can be obtained by multiplying the other by a constant.

Question 14

Given the system $$\begin{cases} x + 2y = -1 \ y - x = 4 \end{cases}$$, is the point $$(-3, 1) a $$solution?

Accepted Answer

Yes, it is a solution to both equations.

Question 15

Which of the following best describes the solution set of the system $$\begin{cases} y = 3x + 6 \ 6x - 2y = -12 \end{cases}$$?

Accepted Answer

The system is consistent and has infinitely many solutions.

Question 16

A company orders 80 promotional items, cups at $$\0.80$ each and bottles at $$\$$1.35$$ each, spending $$\72.80$ in total. Let c$ be the number of cups and b$ the number of bottles. Which system models this situation?

Accepted Answer

$$\begin{cases} c + b = 80 \ 0.80c + 1.35b = 72.80 \end{cases}$$

Question 17

If a system of equations has infinitely many solutions, what must be true about the graphs of the equations?

Accepted Answer

The graphs are the same line.

Systems of Equations in Two Variables: Graphical Solutions and Classification

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