Q: After multiplying the first equation by 2 and the second by 3, what is the resulting system? $$\begin{cases} 3x + 5y = 16 \\ 2x + 4y = 26 \end{cases}$$

$$\begin{cases} 6x + 10y = 32 \\ 6x + 12y = 78 \end{cases}$$

Q: What is the associated matrix $$A$$ for the system $$\begin{cases} 3x + 5y = 16 \\ 2x + 4y = 26 \end{cases}$$?

$$\begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}$$

Question 1

Given the system of equations:

$$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}$$

Which of the following is the correct first step to solve this system using the elimination method?

Accepted Answer

Multiply the first equation by 2 and the second equation by 3, then subtract one from the other to eliminate $$x.$$

Question 2

After multiplying the first equation by 2 and the second by 3, what is the resulting system?

$$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}$$

Accepted Answer

$$\begin{cases} 6x + 10y = 32 \ 6x + 12y = 78 \end{cases}$$

Question 3

Using the elimination method, what is the value of $$y$$ for the system:

$$\begin{cases} 6x + 10y = 32 \ 6x + 12y = 78 \end{cases}$$

Accepted Answer

Question 4

What is the value of $$x$$ after solving for $$y in $$the system:

$$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}

if$$ $$y = 23$$?

Accepted Answer

$$x = -39$$

Question 5

Which matrix operation is used to solve a system of equations using Gaussian elimination?

Accepted Answer

Row reduction

Question 6

If you have the augmented matrix:

$$
\begin{bmatrix}
3 & 5 & | & 16 \
2 & 4 & | & 26
\end{bmatrix}
$$

What is the next step to begin row reduction?

Accepted Answer

Multiply the first row by 2 and the second row by 3.

Question 7

What is the solution to the system $$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}$$?

Accepted Answer

$$x = -39, y = 23$$

Question 8

Which of the following best describes the elimination method for solving systems of equations?

Accepted Answer

Adding or subtracting equations to eliminate one variable.

Question 9

In the context of solving systems, what does 'row reduction' mean?

Accepted Answer

Performing operations to transform the matrix into row-echelon form.

Question 10

If a system of equations has no solution, what is the relationship between the equations?

Accepted Answer

They are inconsistent.

Question 11

Which of the following is a necessary condition for a system of two linear equations to have a unique solution?

Accepted Answer

The equations must have different slopes.

Question 12

What is the associated matrix $$A$$ for the system $$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}$$?

Accepted Answer

$$\begin{bmatrix} 3 & 5 \ 2 & 4 \end{bmatrix}$$

Question 13

If you use row operations to solve the system $$\begin{cases} 3x + 5y = 16 \ 2x + 4y = 26 \end{cases}$$, what is the first row operation you should perform?

Accepted Answer

Multiply the first row by 2 and the second row by 3.

Question 14

Which of the following is NOT a valid row operation in matrix row reduction?

Accepted Answer

Multiplying a row by zero.

Question 15

In a real-world scenario, a chemist is mixing two solutions. The equations $$3x + 5y = 16$$ and $$2x + 4y = 26$$ represent the amounts of two chemicals needed for a reaction. What do the variables $$x$$ and $$y$$ represent?

Accepted Answer

The amounts of the two chemicals being mixed.

Question 16

A medical researcher uses the system $$3x + 5y = 16$$ and $$2x + 4y = 26 to $$determine the dosage of two drugs for a patient. If $$x$$ and $$y$$ are negative, what does this indicate?

Accepted Answer

The dosages are not physically possible.

Question 17

Which of the following best describes the solution to a system of equations in terms of intersection?

Accepted Answer

The solution is the point where the two lines intersect.

Question 18

If you read off the solution from the reduced row-echelon form of the augmented matrix, what do the entries in the last column represent?

Accepted Answer

The solutions for each variable.

Study Notes: Systems of Equations & Matrices (Gaussian Elimination)

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