Q: If $$D = 100$$, $$R = 3$$, and $$T = 2$$, what is the value of $$M$$ using $$M = \frac{D - 2RT}{2R + 2T}$$?

$$\frac{100 - 2 \times 3 \times 2}{2 \times 3 + 2 \times 2} = \frac{100 - 12}{6 + 4} = \frac{88}{10} = 8.8$$

Q: A chemical engineer uses the formula $$D = 2RT + 2RM + 2TM to $$calculate the mass $$M$$ needed for a reaction. If $$D = 50$$, $$R = 4$$, and $$T = 1$$, what is $$M$$?

$$\frac{50 - 2 \times 4 \times 1}{2 \times 4 + 2 \times 1} = \frac{50 - 8}{8 + 2} = \frac{42}{10} = 4.2$$

Q: A medical researcher uses $$D = 2RT + 2RM + 2TM to $$model dosage. If $$D = 60$$, $$R = 5$$, $$T = 2$$, what is $$M$$?

$$\frac{60 - 2 \times 5 \times 2}{2 \times 5 + 2 \times 2} = \frac{60 - 20}{10 + 4} = \frac{40}{14} \approx 2.857$$

Question 1

Which step is NOT necessary when solving $$D = 2RT + 2RM + 2TM$$ for $$M$$?

Accepted Answer

Clear the equation of fractions by multiplying each side by the least common denominator.

Question 2

What is the first algebraic operation performed to isolate $$M in$$ $$D = 2RT + 2RM + 2TM$$?

Accepted Answer

Subtract $$2RT$$ from both sides.

Question 3

After subtracting $$2RT$$ from both sides of $$D = 2RT + 2RM + 2TM$$, what is the resulting equation?

Accepted Answer

$$D - 2RT = 2RM + 2TM$$

Question 4

Which property is used to combine $$2RM + 2TM$$ into $$(2R + 2T)M$$?

Accepted Answer

Distributive property

Question 5

What is the final expression for $$M$$ after isolating it in $$D = 2RT + 2RM + 2TM$$?

Accepted Answer

$$\frac{D - 2RT}{2R + 2T}$$

Question 6

If $$D = 100$$, $$R = 3$$, and $$T = 2$$, what is the value of $$M$$ using $$M = \frac{D - 2RT}{2R + 2T}$$?

Accepted Answer

$$\frac{100 - 2 	imes 3 	imes 2}{2 	imes 3 + 2 	imes 2} = \frac{100 - 12}{6 + 4} = \frac{88}{10} = 8.8$$

Question 7

Which of the following is an equivalent form of $$M = \frac{D - 2RT}{2R + 2T}$$?

Accepted Answer

$$M = \frac{D}{2(R + T)} - \frac{RT}{R + T}$$

Question 8

In the context of solving for a specified variable, what is the purpose of the multiplication property of equality?

Accepted Answer

To isolate the specified variable by dividing or multiplying both sides by the same value.

Question 9

If $$D = 2RT + 2RM + 2TM$$ and you mistakenly divide both sides by $$2R$$ instead of $$2R + 2T$$, what is the resulting expression for $$M$$?

Accepted Answer

$$\frac{D - 2RT}{2R}$$

Question 10

A chemical engineer uses the formula $$D = 2RT + 2RM + 2TM to $$calculate the mass $$M$$ needed for a reaction. If $$D = 50$$, $$R = 4$$, and $$T = 1$$, what is $$M$$?

Accepted Answer

$$\frac{50 - 2 	imes 4 	imes 1}{2 	imes 4 + 2 	imes 1} = \frac{50 - 8}{8 + 2} = \frac{42}{10} = 4.2$$

Question 11

Which step would you use if the equation contained fractions?

Accepted Answer

Clear the equation of fractions by multiplying each side by the least common denominator.

Question 12

If $$D = 2RT + 2RM + 2TM$$ and $$M$$ must be isolated, which property allows you to rewrite $$2RM + 2TM as$$ $$(2R + 2T)M$$?

Accepted Answer

Distributive property

Question 13

A medical researcher uses $$D = 2RT + 2RM + 2TM to $$model dosage. If $$D = 60$$, $$R = 5$$, $$T = 2$$, what is $$M$$?

Accepted Answer

$$\frac{60 - 2 	imes 5 	imes 2}{2 	imes 5 + 2 	imes 2} = \frac{60 - 20}{10 + 4} = \frac{40}{14} \approx 2.857$$

Question 14

If you incorrectly combine $$2RM + 2TM as$$ $$2(RM + TM)$$, what would be the next step to isolate $$M$$?

Accepted Answer

Divide both sides by $$2(RM + TM)$$

Question 15

Which step is used to rewrite the equation so that all terms containing the specified variable are on one side?

Accepted Answer

Use the addition property of equality

Question 16

If $$D = 2RT + 2RM + 2TM$$ and you want to solve for $$R$$, what would be your first step?

Accepted Answer

Group all terms containing $$R on $$one side of the equation.

Question 17

If $$D = 2RT + 2RM + 2TM$$ and $$D = 0$$, $$R = 2$$, $$T = 3$$, what is $$M$$?

Accepted Answer

$$\frac{0 - 2 	imes 2 	imes 3}{2 	imes 2 + 2 	imes 3} = \frac{0 - 12}{4 + 6} = \frac{-12}{10} = -1.2$$

Question 18

Which step is skipped when there are no fractions or grouping symbols in the equation?

Accepted Answer

Clear the equation of fractions by multiplying each side by the least common denominator.

Solving Equations for a Specified Variable: Isolating M

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