Q: Which of the following is the correct determinant for the coefficient matrix of the system: $$\begin{cases} 2x + 3y = 7 \\ 4x - y = 5 \end{cases}$$

$$\begin{vmatrix} 2 & 3 \\ 4 & -1 \end{vmatrix} = (2)(-1) - (4)(3) = -2 - 12 = -14$$

Q: Which of the following systems is inconsistent?

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

Question 1

Given the system of equations:

$$\begin{cases} -4x + 2y = -20 \ x + 2y = 3 \end{cases}$$

Which of the following is the correct solution for $$(x, y)$$ using the elimination method?

Accepted Answer

$$(x, y) = (-5, 4)$$

Question 2

Which of the following statements about Cramer's Rule is TRUE?

Accepted Answer

Cramer's Rule can only be used if the determinant of the coefficient matrix is nonzero.

Question 3

Evaluate the expression $$2x_1 + 3x_2 - x_3$ for x_1$ = 1$$, $$x_2 = 2$, and x_3$ = 3.$$

Accepted Answer

$$4$$

Question 4

If the determinant of the coefficient matrix in a $$2 	imes 2$$ system is zero, what does this imply about the system?

Accepted Answer

The system has no solution or infinitely many solutions.

Question 5

A chemist is mixing two solutions. The first solution contains $$x$$ grams of chemical A and the second contains $$y$$ grams of chemical B. If $$2x + y = 10$$ and $$x - y = 1$$, how many grams of each chemical are in the mixture?

Accepted Answer

$$x = 3.67, y = 2.33$$

Question 6

Which of the following is the correct determinant for the coefficient matrix of the system:

$$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$

Accepted Answer

$$\begin{vmatrix} 2 & 3 \ 4 & -1 \end{vmatrix} = (2)(-1) - (4)(3) = -2 - 12 = -14$$

Question 7

Suppose you are given the system:

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

What can you conclude about the solutions to this system?

Accepted Answer

There are infinitely many solutions.

Question 8

A business has two types of products, A and B. Each unit of A requires 2 hours of labor and 1 hour of machine time. Each unit of B requires 1 hour of labor and 3 hours of machine time. If the business has 8 hours of labor and 10 hours of machine time available, how many units of each product can be produced if all resources are used?

Accepted Answer

3 units of A and 2 units of B

Question 9

Which of the following is NOT a valid step in solving a system of equations using the substitution method?

Accepted Answer

Add the two equations together to eliminate a variable.

Question 10

Given the system:

$$\begin{cases} 3x - 2y = 7 \ 2x + y = 4 \end{cases}$$

What is the value of $$y$$?

Accepted Answer

$$y = 2$$

Question 11

If a system of equations has no solution, what is the geometric interpretation of the lines represented by the equations?

Accepted Answer

The lines are parallel.

Question 12

Which of the following systems is inconsistent?

Accepted Answer

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

Question 13

A medical researcher is analyzing the relationship between two drugs. The total dosage must be 100 mg, and the dosage of drug A must be twice that of drug B. What are the dosages of drugs A and B?

Accepted Answer

Drug A: 66.67 mg, Drug B: 33.33 mg

Question 14

Which of the following best describes the solution to the system:

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

Accepted Answer

There are infinitely many solutions.

Question 15

If $$x = 2$$ and $$y = -1$$, what is the value of $$3x + 4y$$?

Accepted Answer

$$-2$$

Solving Systems of Equations Using Cramer's Rule and Substitution

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