Q: A chemist wants to create 70 oz of a solution that is 50% alcohol by mixing a solution that is 40% alcohol with one that is 75% alcohol. Which system of equations correctly models this situation if $$x is $$the amount (in oz) of 40% solution and $$y is $$the amount (in oz) of 75% solution?

$$\begin{cases} x + y = 70 \\ 0.4x + 0.75y = 0.5 \times 70 \end{cases}$$

Q: Which of the following systems has no solution?

$$\begin{cases} x + y = 2 \\ 2x + 2y = 5 \end{cases}$$

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

$$\begin{bmatrix} 1 & 2 & | & 5 \\ 3 & -1 & | & 4 \end{bmatrix}$$

Question 1

A chemist wants to create 70 oz of a solution that is 50% alcohol by mixing a solution that is 40% alcohol with one that is 75% alcohol. Which system of equations correctly models this situation if $$x is $$the amount (in oz) of 40% solution and $$y is $$the amount (in oz) of 75% solution?

Accepted Answer

$$\begin{cases} x + y = 70 \ 0.4x + 0.75y = 0.5 	imes 70 \end{cases}$$

Question 2

Given the system of equations:

$$\begin{cases} 2x + y + z = 7 \ 4x + 2y - 3z = -20 \end{cases}$$

Which of the following is the correct first row operation to eliminate $$y$$ from the second equation?

Accepted Answer

Subtract twice the first equation from the second: $$R_2$$ \leftarrow $$R_2 - 2R_1$

Question 3

Suppose you have a mixture that is 40% alcohol and another that is 75% alcohol. You want to make 70 oz of a 50% alcohol solution. How much of the 75% alcohol mixture is required?

Accepted Answer

$$20 oz$$

Question 4

Which of the following best describes the process of using matrices and row operations to solve a system of equations?

Accepted Answer

Transforming the system into an augmented matrix and using row operations to reach row-echelon form

Question 5

A pharmacist needs to mix a solution that is 30% saline with one that is 80% saline to obtain 100 oz of a 50% saline solution. Which equation represents the total amount of saline in the final mixture if $$x is $$the amount of 30% solution and $$y is $$the amount of 80% solution?

Accepted Answer

$$0.3x + 0.8y = 50$$

Question 6

Given the system:

$$\begin{cases} 2x + y + z = 7 \ 4x + 2y - 3z = -20 \end{cases}$$

After performing $$R_2$$ \leftarrow $$R_2 - 2R_1$, what is the new second equation?

Accepted Answer

$$0x + 0y - 5z = -34$$

Question 7

If a solution to a system of equations is $$(x, y, z) = (2, 1, 2)$$, which of the following must be true?

Accepted Answer

Substituting $$x=2$$, $$y=1$$, $$z=2$$ into each equation yields a true statement

Question 8

A biologist is preparing a nutrient solution by mixing two solutions: one with 10% nutrient and one with 60% nutrient. She needs 200 oz of a 25% nutrient solution. How much of the 10% solution should she use?

Accepted Answer

$$140 oz$$

Question 9

Which of the following is NOT a valid row operation when solving a system using matrices?

Accepted Answer

Multiplying a row by zero

Question 10

A solution is made by mixing $$x oz of a 30$$% acid solution with $$y oz of a 70$$% acid solution to obtain 100 oz of a 50% acid solution. What is the value of $$x$$?

Accepted Answer

$$25$$

Question 11

Which of the following systems has no solution?

Accepted Answer

$$\begin{cases} x + y = 2 \ 2x + 2y = 5 \end{cases}$$

Question 12

A medical technician needs to prepare 500 mL of a 12% saline solution by mixing a 10% solution and a 20% solution. How much of the 20% solution should be used?

Accepted Answer

$$100$$ 	ext{ mL}$

Question 13

Which of the following best describes a consistent and dependent system?

Accepted Answer

A system with infinitely many solutions

Question 14

If you have the augmented matrix:

$$\begin{bmatrix} 1 & 2 & | & 5 \ 2 & 4 & | & 10 \end{bmatrix}$$

What does this matrix represent?

Accepted Answer

A system with infinitely many solutions

Question 15

A solution is made by mixing 30 oz of a 20% sugar solution with 70 oz of a 60% sugar solution. What is the concentration of sugar in the final mixture?

Accepted Answer

$$48\%$$

Question 16

Which of the following is the correct augmented matrix for the system:

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

Accepted Answer

$$\begin{bmatrix} 1 & 2 & | & 5 \ 3 & -1 & | & 4 \end{bmatrix}$$

Question 17

A chemist mixes $$x$$ liters of a 10% acid solution with $$y$$ liters of a 30% acid solution to obtain 20 liters of a 25% acid solution. What is the value of $$y$$?

Accepted Answer

$$15$$

Question 18

Which of the following is a valid step in solving a system using Gaussian elimination?

Accepted Answer

Adding a multiple of one row to another row

Question 19

A solution is made by mixing 100 mL of a 25% saline solution with 200 mL of a 40% saline solution. What is the concentration of saline in the final mixture?

Accepted Answer

$$35\%$$

Systems of Equations and Mixture Problems: Precalculus Study Notes

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