Equations - High School Math

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Question

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?

Answer

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:

hours.

If Huck works hours and Tom works hours, he works more hours than Tom.

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Question

Simplify the fraction to the lowest terms:

$\frac{924}{1092}$

Answer

Find the common multiple between the numerator and denominator.

$\frac{924}{1092}$

divide numerator and denominator by 3:

$\frac{308}{364}$

divide numerator and denominator by 7:

$\frac{44}{52}$

divide numerator and denominator by 4:

$\frac{11}{13}$

Cannot be divided any more- lowest terms.

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Question

If given the equation , with a positive integer, the result must be an integer multiple of:

Answer

The mathematical expression given in the question is . Adding together like terms, , this can be simplified to . The expression can be factored as . For every positive integer , must be a multiple of 5. If , then , which is not an integer multiple of 2, 8, 10, or 15. Therefore, the correct answer is 5.

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Question

Solve the following equation for x in terms of the other variables:

$\frac{ax}{b-ax}=2$

Answer

$\frac{ax}{b-ax}=2$

Multiply both sides by to get:

Distribute the :

Combine like terms:

Divide both sides by :

$x=\frac{2b}{3a}$

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Question

Solve the following equation for x in terms of the other variables:

Answer

Divide both sides by :

$x=\frac{a}{bc}$

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Question

Solve the system of equations.

Answer

Isolate in the first equation.

Plug into the second equation to solve for .

Plug into the first equation to solve for .

Now we have both the and values and can express them as a point: .

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Question

Solve for and .

Answer

1st equation:

2nd equation:

Subtract the 2nd equation from the 1st equation to eliminate the "2y" from both equations and get an answer for x:

Plug the value of into either equation and solve for :

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Question

Cindy's Cotton Candy sells cotton candy by the bag. Her monthly fixed costs are . It costs to make each bag and she sells them for .

What is the monthly break-even point?

Answer

The break-even point occurs when the .

The equation to solve becomes

so the break-even point is .

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Question

Cindy's Cotton Candy sells cotton candy by the bag. Her monthly fixed costs are . It costs to make each bag and she sells them for .

To make a profit of , how many bags of cotton candy must be sold?

Answer

So the equation to solve becomes , or must be sold to make a profit of .

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Question

Solve for and to satisfy both equations in the system:

$\begin{Bmatrix} 6x+y = 25 \ 2x-3y = 25 \end{Bmatrix}$

Answer

The two equations in this system can be combined by addition or subtraction to solve for and . Isolate the variable to solve for it by multiplying the top equation by so that when the equations are combined the term disappears.

$3\times(6x + y = 25)$

$\rightarrow 18x+3y = 75$

$\rightarrow (18x +3y = 75) + (2x - 3y = 25)$

$\rightarrow 20x = 100$

Divide both sides by to find as the value for .

Substituting for in both of the two equations in the system and solving for gives a value of for .

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Question

Solve for :

$\left | 12x - 39 \right | = 60$

Answer

Rewrite $\left | 12x - 39 \right | = 60$ as a compound statement and solve each part separately:

$\left | 12x - 39 \right | = 60$

$12x - 39 = - 60 \textrm{ or }12x - 39 = 60$

$12x \div 12 = - 21 \div 12$

$x = -1 \frac{3}{4}$

$12x \div 12 = 99 \div 12$

$x = 8 \frac{1}{4}$

The solution set is $\left { -1 \frac{3}{4} ,8 \frac{1}{4}\right }$

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Question

Solve for :

$\left | 5x + 39 \right | = 44$

Answer

Rewrite $\left | 5x + 39 \right | = 44$ as a compound statement and solve each part separately:

$\left | 5x + 39 \right | = 44$

$5x + 39 = - 44 \textrm{ or }5x + 39 = 44$

$5x \div 5 = - 83 \div 5$

$x = -16\frac{3}{5}$

$5x \div 5 =5 \div 5$

Therefore the solution set is $\left {-16\frac{3}{5}, 1 \right }$ .

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Question

Solve the following equation for .

Answer

The first step in solving this equation is to distribute the 2 through the parentheses. This gives us:

$2\cdot z + 2\cdot 3 = 18$

Next, we subtract 6 from both sides, in order to get the variable alone on one side of the equation:

Finally, we divide both sides by 2 to solve for :

$\frac{2z}{2} = \frac{12}{2}$

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Question

Solve the following equation for :

Answer

The first step in solving this equation is to combine the like terms. That is, move all of the terms with an in it to one side of the equation, and all the terms without an to the other side. Let's begin with moving the terms to the left side of the equation. We do this by subtracting from each side:

Next, we combine the terms without an on the right side. We do this by subtracting 3 from both sides:

Finally, we divide both sides of the equation by 2 to solve for :

$\frac{2x}{2} = \frac{-10}{2}$

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Question

Solve the following equation for :

Answer

The first step to solve this equation is expand out the right-hand side of the equation by multiplying the 4 through the parentheses:

$9x + 3x + 2 + 10 = 4\cdot 2+4\cdot x$

Next, we combine like terms. That is, we put all of the terms with an on one side of the equation and all the terms without on the other. We'll start by subtracting from both sides:

And now we'll move the non- terms to the right-hand side by subtracting 2 and 10 from both sides:

Combining these terms, this simplifies to:

Next, we divide both sides by 8:

$\frac{8x}{8}=\frac{-4}{8}$

Which gives:

$x = -\frac{1}{2}$

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Question

Solve the following equation for :

Answer

The first step in solving this equation is to distribute the 3 and the 4 through the parentheses:

$3\cdot x+3\cdot2 = 4\cdot x-4\cdot 3$

Simplify:

Now, we want to get like terms on the same sides of the equation. That is, all of the terms with an should be on one side, and those without an should be on the other. To do this, we first subtract from both sides:

Simplify:

Now, we subtract 6 from both sides:

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Question

Solve the following equation for y:

$\frac{y}{2} + 3 = -7$

Answer

$\frac{y}{2} + 3 = -7$

The first step in solving this equation it to subtract 3 from both sides, so that the term with is alone on one side of the equals sign.

$\frac{y}{2} + 3 -3 = -7-3$

which simplifies to:

$\frac{y}{2} = -10$

From here, we multiply both sides by 2:

$2\cdot \frac{y}{2} = -10\cdot 2$

This gives:

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Question

Solve the following equation for :

Answer

The first step to solving this equation is to combine the terms with in them on the left-hand side of the equation. This gives:

Next, we can subtract from both sides of the equation:

Which simplifies to:

Then, we subtract 2 from both sides of the equation:

And to finish, we now divide both sides by 2:

$\frac{2x}{2}=\frac{4}{2}$

Which simplifies to:

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Question

Solve the following equation for :

$\frac{x}{3}+3(x+2)=7+x$

Answer

$\frac{x}{3}+3(x+2)=7+x$

The first step in solving this equation is to distribute the 3 through the parentheses on the left-hand side of the equation:

$\frac{x}{3}+3\cdot x+3\cdot 2=7+x$

Which gives:

$\frac{x}{3}+3x+6=7+x$

Now, in order to get rid of the fraction, we can multiply the whole equation by 3:

We can now combine the like terms on the left-hand side, which gives:

Now, we subtract from both sides:

Next, we subtract 18 from both sides:

Last, we divide both sides by 7 to solve for :

$\frac{7x}{7}=\frac{3}{7}$

Which gives,

$x=\frac{3}{7}$

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Question

For which of the following functions is the result of a positive integer?

Answer

Simply plugging in -2 into each answer choice will determine the correct answer:

The key to solving this problem is to remember the order of operations and that negative numbers squared are positive, while negative numbers raised to the third power are negative.

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