The general equation of a circle is \(\displaystyle (x - h)^{2} + (y - k)^{2} = r^{2}\), where the center of the circle is \(\displaystyle (h, k)\) and the radius is \(\displaystyle r\).

Thus, we plug the values given into the above equation to get \(\displaystyle (x - 2)^{2} + (y - 4)^{2} = 100\). 

The standard formula for a circle is \(\displaystyle (x-a)^2+(y-b)^2=r^2\), with \(\displaystyle (a,b)\) the center of the circle and \(\displaystyle r\) the radius.

Plug in our given information.

\(\displaystyle (x-a)^2+(y-b^2)=r^2\)

\(\displaystyle (x-2)^2+(y-3)^2=25\)

This describes what we are looking for.  This equation is not one of the answer choices, however, so subtract \(\displaystyle 25\) from both sides.

\(\displaystyle (x-2)^2+(y-3)^2-25=0\)

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 \(\displaystyle y\).