Tom paints for a total of \(\displaystyle 4\) hours (2 on his own, 2 with Huck's help). Since he paints at a rate of \(\displaystyle 5\) feet per hour, use the formula

\(\displaystyle distance = rate \times time\) (or \(\displaystyle d = rt\))

to determine the total length of the fence Tom paints.

\(\displaystyle d = (5)(4)\)

\(\displaystyle d = 20\) feet

Subtracting this from the total length of the fence \(\displaystyle 100\) feet gives the length of the fence Tom will NOT paint: \(\displaystyle 100 - 20 = 80\) feet. If Huck finishes the job, he will paint that \(\displaystyle 80\) feet of the fence. Using \(\displaystyle d = rt\), we can determine how long this will take Huck to do:

\(\displaystyle 80 = 8(t)\)

\(\displaystyle t = 10\) hours.

If Huck works \(\displaystyle 10\) hours and Tom works \(\displaystyle 4\) hours, he works \(\displaystyle 6\) more hours than Tom.

Find the common multiple between the numerator and denominator.

\(\displaystyle \frac{924}{1092}\)

divide numerator and denominator by 3:

\(\displaystyle \frac{308}{364}\)

divide numerator and denominator by 7:

\(\displaystyle \frac{44}{52}\)

divide numerator and denominator by 4:

\(\displaystyle \frac{11}{13}\)

Cannot be divided any more- lowest terms.

\(\displaystyle \frac{ax}{b-ax}=2\)  

Multiply both sides by \(\displaystyle (b-ax)\) to get:

\(\displaystyle ax=2(b-ax)\)

Distribute the \(\displaystyle 2\):

\(\displaystyle ax=2b-2ax\) 

Combine like terms:

\(\displaystyle 3ax=2b\)

Divide both sides by \(\displaystyle 3a\):

\(\displaystyle x=\frac{2b}{3a}\)

\(\displaystyle bcx=a\)

Divide both sides by \(\displaystyle bc\):

\(\displaystyle x=\frac{a}{bc}\)

The mathematical expression given in the question is \(\displaystyle 8x + 2x -5\). Adding together like terms, \(\displaystyle 8x + 2x\), this can be simplified to \(\displaystyle 10x -5\). The expression \(\displaystyle 10x -5\) can be factored as \(\displaystyle 5(2x -1)\). For every positive integer \(\displaystyle x\), \(\displaystyle 5(2x -1)\) must be a multiple of 5. If \(\displaystyle x=1\), then \(\displaystyle 5(2x - 1) = 5\), which is not an integer multiple of 2, 8, 10, or 15. Therefore, the correct answer is 5.

\(\displaystyle Costs=150+2.50x\)

\(\displaystyle Revenues = 4.00x\)

The break-even point occurs when the \(\displaystyle costs=revenues\).

The equation to solve becomes

\(\displaystyle 4x=150+2.5x\) so the break-even point is \(\displaystyle 100 \; bags\).

\(\displaystyle Costs=150+2.50x\)

\(\displaystyle Revenues = 4.00x\)

\(\displaystyle Profits = Revenues - Costs = 4x - (150+2.5x)=1.5x -150\)

So the equation to solve becomes \(\displaystyle 150 = 1.5x - 150\), or \(\displaystyle x=200\; bags\) must be sold to make a profit of \(\displaystyle \$150\).

The two equations in this system can be combined by addition or subtraction to solve for \(\displaystyle x\) and \(\displaystyle y\). Isolate the \(\displaystyle x\) variable to solve for it by multiplying the top equation by \(\displaystyle 3\) so that when the equations are combined the \(\displaystyle y\) term disappears. 

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

\(\displaystyle \rightarrow 18x+3y = 75\)

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

\(\displaystyle \rightarrow 20x = 100\)

Divide both sides by \(\displaystyle 20\) to find \(\displaystyle 5\) as the value for \(\displaystyle x\).

Substituting \(\displaystyle 5\) for \(\displaystyle x\) in both of the two equations in the system and solving for \(\displaystyle y\) gives a value of \(\displaystyle -5\) for \(\displaystyle y\). 

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

\(\displaystyle \left | 12x - 39 \right | = 60\)

\(\displaystyle 12x - 39 = - 60 \textrm{ or }12x - 39 = 60\)

\(\displaystyle 12x - 39 = - 60\)

\(\displaystyle 12x - 39 + 39 = - 60+ 39\)

\(\displaystyle 12x = - 21\)

\(\displaystyle 12x \div 12 = - 21 \div 12\)

\(\displaystyle x = -1 \frac{3}{4}\)

\(\displaystyle 12x - 39 = 60\)

\(\displaystyle 12x - 39 + 39 = 60+ 39\)

\(\displaystyle 12x = 99\)

\(\displaystyle 12x \div 12 = 99 \div 12\)

\(\displaystyle x = 8 \frac{1}{4}\)

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

First, notice that we can factor \(\displaystyle a^2-b^2\) into the form (a-b)(a+b). We are told that a-b=3, so we can substitute that into the first equation.

\(\displaystyle (a-b)(a+b)=33\)

\(\displaystyle 3(a+b)=33\)

If we divide both sides by 3, we can obtain the value of a+b.

\(\displaystyle a+b=11\)

We now have a system of equations: a-b = 3, a+b = 11. We will solve this system by elimination. If we add the two equations together, we obtain the following:

\(\displaystyle 2a=14\).

Divide both sides by 2.

\(\displaystyle a=7.\)

Going back to the equation a-b = 7, we can solve for b.

\(\displaystyle 7-b=3\)

Add b to both sides.

\(\displaystyle 7=3+b\)

Subtract 3 from both sides.

\(\displaystyle b=4.\)

Ultimately, the question asks us to determine the value of \(\displaystyle a^2+b^2\).

\(\displaystyle a^2+b^2\) = \(\displaystyle 7^{2}+4^{2}=49+16=65\).

The answer is 65.