Algebra II - Basic Single-Variable Algebra

Find the solution to the following equation.

Explanation

Find the solution to the following equation.

To find the solution of this equation, we need to find the value for x, which makes the given equation true.

We do this by working backwards and manipulating the equation.

The easiest first step would probably be to subtract 23 from both sides:

By doing so, we can simplify the equation to get:

Next, simply divide both sides by 9 to get the answer!

$9x\div 9=36\div9$

So we get:

Solve the equation:

$\frac{2}{5}$

$\frac{2}{13}$

$-\frac{2}{13}$

$\frac{20}{13}$

Explanation

Add on both sides.

Combine like-terms on both sides.

Subtract nine from both sides.

Divide by five on both sides.

$\frac{5x}{5}=\frac{2}{5}$

The answer is: $\frac{2}{5}$

A large tub has two faucets. The hot water faucet, if turned all the way up, can fill the tub in 16 minutes; the cold water faucet, in 12 minutes. Which of the following choices comes closest to the amount of time it takes for both faucets working together to fill the tub?

$7 \textup{ min}$

$8 \textup{ min}$

$5 \textup{ min}$

$6 \textup{ min}$

$4\textup{ min}$

Explanation

Work problems can be solved by looking at them as rate problems.

The hot faucet can fill up the tub at a rate of 16 minutes per tub, or $\frac{1}{16}$ tub per minute. The cold faucet, similarly, can fill up the tub at a rate of 12 minutes per tub, or $\frac{1}{12}$ tub per minute.

Suppose the tub fills up in minutes. Then, at the end of this time, the hot faucet has filled up $\frac{1}{16} X$ tub, and the cold faucet has filled up $\frac{1}{12} X$ tub, for a total of one tub. We can set up this equation and solve for :

$\frac{1}{16} X + \frac{1}{12} X = 1$

$48 \cdot \left (\frac{1}{16} X + \frac{1}{12} X \right )= 48 \cdot 1$

$48 \cdot \frac{1}{16} X + 48 \cdot \frac{1}{12} X = 48$

$7X\div 7 = 48 \div 7$

$X \approx 6.86$

Of the given choices, 7 minutes comes closest.

Solve for :

$(\frac{1}{2})x +7=15$

Explanation

$(\frac{1}{2})x=15-7=8$

$x=(\frac{2}{1})\times8=16$

Set up the equation: Nine more than three times the cube of a number is four.

Explanation

Split the sentence into parts.

The cube of a number:

Three times the cube of a number:

Nine more than three times the cube of a number:

Is four:

Combine the parts to form the equation.

The answer is:

Sarah notices her map has a scale of $\frac{1}{4}; in=1; mile$ . She measures between Beaver Falls and Chipmonk Cove. How far apart are the cities?

Explanation

$\frac{1}{4}; in=1; mile$ is the same as

So to find out the distance between the cities

$12.5; in \cdot \frac{4; miles}{1;in }=50; miles$

varies inversely with and the square root of . When and , $y=\frac{2}{9}$ . Find when and .

$\small \frac{1}{2}$

$\small \frac{1}{4}$

None of these answers are correct

Explanation

First, we can create an equation of variation from the the relationships given:

$y = k \cdot \frac{1}{x \cdot \sqrt{z}}$

Next, we substitute the values given in the first scenario to solve for $\small k$ :

$\frac{2}{9} = k \cdot \frac{1}{4 \cdot \sqrt{81}} = k \cdot \frac{1}{4 \cdot 9} = k \cdot \frac{1}{36}$

$\small \rightarrow k = 36 \cdot \frac{2}{9} = \frac{36}{9} \cdot 2 = 4 \cdot 2 = 8$

Using the value for , we can now use the second values for and to solve for :

$\small Y = 8 \cdot \frac{1}{8 \cdot \sqrt{4}} = 8 \cdot \frac{1}{8 \cdot 2} = 8 \cdot \frac{1}{16} = \frac{1}{2}$

Solve the equation:

$\frac{2}{5}$

$\frac{2}{13}$

$-\frac{2}{13}$

$\frac{20}{13}$

Explanation

Add on both sides.

Combine like-terms on both sides.

Subtract nine from both sides.

Divide by five on both sides.

$\frac{5x}{5}=\frac{2}{5}$

The answer is: $\frac{2}{5}$

Tom is painting a fence feet long. He starts at the West end of the fence and paints at a rate of feet per hour. After hours, Huck joins Tom and begins painting from the East end of the fence at a rate of feet per hour. After hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

$6\ hours$

$10\ hours$

$4\ hours$

$3\ hours$

$5\ hours$

Explanation

Tom paints for a total of hours (2 on his own, 2 with Huck's help). Since he paints at a rate of feet per hour, use the formula

$distance = rate \times time$ (or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence feet gives the length of the fence Tom will NOT paint: feet. If Huck finishes the job, he will paint that feet of the fence. Using , we can determine how long this will take Huck to do: