Q: Sonya has 30 coins in dimes and quarters totaling $$5.70. Which system of equations models this scenario if d$ is the number of dimes and q$ is the number of quarters?

$$\begin{cases} d + q = 30 \\ 0.10d + 0.25q = 5.70 \end{cases}$$

Q: What is the Reduced Row Echelon Form (RREF) of the matrix $$\begin{bmatrix} 1 & 2 & 8 \\ 1 & 1 & 6 \end{bmatrix}$$?

$$\begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 2 \end{bmatrix}$$

Q: What is the augmented matrix for the system $$2x + 3y = 10$$ and $$2x + y = 5$$?

$$\begin{bmatrix} 2 & 3 & 10 \\ 2 & 1 & 5 \end{bmatrix}$$

Question 1

Given the system of equations $$y = 2x - 1$$ and $$y = -x + 8$$, what is the solution to the system?

Accepted Answer

$$(x, y) = (3, 5)$$

Question 2

Which method is most efficient for solving a system of equations in standard form, such as $$2x + 3y = 12$$ and $$3x + 8y = 24$$?

Accepted Answer

Elimination Method

Question 3

What does the solution to a system of linear equations represent graphically?

Accepted Answer

The intersection point of the lines

Question 4

Sonya has 30 coins in dimes and quarters totaling $$5.70. Which system of equations models this scenario if d$ is the number of dimes and q$ is the number of quarters?

Accepted Answer

$$\begin{cases} d + q = 30 \ 0.10d + 0.25q = 5.70 \end{cases}$$

Question 5

If Texas Instruments has a fixed monthly overhead of $$24,500 and each calculator costs 10$ to make and sells for 45$, which equation represents the break-even point for x$ calculators?

Accepted Answer

$$45x = 10x + 24,500$$

Question 6

What is the Reduced Row Echelon Form (RREF) of the matrix $$\begin{bmatrix} 1 & 2 & 8 \ 1 & 1 & 6 \end{bmatrix}$$?

Accepted Answer

$$\begin{bmatrix} 1 & 0 & 4 \ 0 & 1 & 2 \end{bmatrix}$$

Question 7

Which calculator command is used to find the Reduced Row Echelon Form (RREF) of a matrix on the TI-84?

Accepted Answer

rref(

Question 8

A system of three linear equations in three variables represents what geometric objects in three-dimensional space?

Accepted Answer

Planes

Question 9

What is the solution to the system $$x + 2y = 8$$ and $$x + y = 6$$?

Accepted Answer

$$(x, y) = (4, 2)$$

Question 10

Which of the following is a valid equation manipulation when using the elimination method?

Accepted Answer

Adding a non-zero multiple of one equation to another

Question 11

If the solution to a system of equations is the point where the lines intersect, what does it mean if the lines are parallel?

Accepted Answer

There is no solution

Question 12

Which of the following systems is best solved by the substitution method?

Accepted Answer

$$y = 3x + 2$$ and $$y = -x + 5$$

Question 13

What is the augmented matrix for the system $$2x + 3y = 10$$ and $$2x + y = 5$$?

Accepted Answer

$$\begin{bmatrix} 2 & 3 & 10 \ 2 & 1 & 5 \end{bmatrix}$$

Question 14

If a system of equations has infinitely many solutions, what does this mean about the graphs of the equations?

Accepted Answer

The lines are coincident (the same line)

Question 15

Which of the following best describes an augmented matrix?

Accepted Answer

A rectangular array of the numbers in a system of linear equations

Question 16

What is the first step in solving a system by the elimination method?

Accepted Answer

Manipulate the equations to eliminate one variable

Question 17

If the RREF of a $$3 	imes 4$$ augmented matrix is $$\begin{bmatrix} 1 & 0 & 0 & 2 \ 0 & 1 & 0 & -1 \ 0 & 0 & 1 & 3 \end{bmatrix}$$, what is the solution to the system?

Accepted Answer

$$(x, y, z) = (2, -1, 3)$$

Question 18

Which of the following is a real-world application of systems of linear equations?

Accepted Answer

Determining the break-even point for a business

Question 19

When using the TI-84 calculator to graph two lines, which sequence of buttons allows you to enter the equations?

Accepted Answer

[y=]

Question 20

If you have the system $$x + y + z = 6$$, $$2x - y + z = 3$$, and $$x + 2y - z = 4$$, what geometric object does the solution represent?

Accepted Answer

The intersection point of three planes

Systems of Linear Equations and Matrices – College Algebra Study Notes

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