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Quiz 4
1. Solve the equation $3x+2=2(x-4)$.
Solution: First distribute the $2$ on the right-hand side to get
$$3x+2=2x-8.$$
Now subtract $2x$ to get
$$x+2=-8.$$
Subtract $2$ to get
$$x=-10.$$
2. Solve $A = \dfrac{2B}{C}$ for $B$.
Solution: Multiply by $C$ to get
$$AC = 2B.$$
Now divide by $2$ to get
$$\dfrac{AC}{2}=B.$$
3. Solve the formula $S = \dfrac{3T}{2W-5X}$ for $W$.
Solution: Take the reciprocal of each side to get
$$\dfrac{1}{S} = \dfrac{2W-5X}{3T}.$$
Now multiply both sides by $3T$ to get
$$\dfrac{3T}{S} = 2W-5X.$$
Add $5X$ to get
$$\dfrac{3T}{S} + 5X = 2W.$$
Now divide both sides by $2$ to get
$$\dfrac{3T}{2S} + \dfrac{5X}{2}=W.$$
4. Bob is 3 years older than Mary. Together the sum of their ages is 43. How old is each?
Solution: Let $B$ denote the age of Bob and let $M$ denote the age of Mary. To say "Bob is 3 years older than Mary" means, in symbols, $B+3=M$ and to say "the sum of their ages is 43" means, in symbols, $B+M=43$. This gives us the following system of equations:
$$\left\{\begin{array}{ll}
B+3=M & (i)\\
B+M=43& (ii).
\end{array}\right.$$
Plugging in the value of $M$ determined by equation $(i)$ into equation $(ii)$ yields the equation
$$B+(B+3)=43.$$
Simplify the left side to get
$$2B+3=43.$$
Subtract $3$ from both sides to get
$$2B=40.$$
Now divide both sides by $2$ to get
$$B=\dfrac{40}{2}=20.$$
This tells us that Bob is 20 years old. To find Mary's age, we use equation (i) to see that $M=20+3=23$, i.e., Mary is 23 years old.
5. In a triangle, the measure of the first angle is 2 times the measure of the second. The measure of the third is $80^{\circ}$ more than the measure of the second. Use the fact that the sum of the three angles in a triangle is $180^{\circ}$ to find the measure of each angle.
Solution: Let $x$ denote the measure of the second angle. Then the first angle's measure is $2x$ and the third angle's measure is $x+80^{\circ}$. Using the fact that the three sum to $180^{\circ}$ gives us the equation
$$2x+x+(x+80^{\circ})=180^{\circ}.$$
Simplify the left side to get
$$4x+80^{\circ}=180^{\circ}.$$
Substract $80^{\circ}$ from both sides to get
$$4x=100^{\circ}.$$
Now divide both sides by $3$ to conclude
$$x = 25^{\circ}.$$
This means that the second angle measures $25$ degrees. Hence the first angle measures $50$ degrees and the third angle measures $25+80=105$ degrees.
