Step 1: The least common denominator of 4 and 3 is 12.  Multiply each term by this LCD:

\(\displaystyle 12\left ( \frac{w+2}{4} \right )+12\left ( \frac{w}{3}\right )=6(12)\)

Step 2: Simplify the fractions:

\(\displaystyle 3w+6+4w=72\)

Step 3: Isolate \(\displaystyle w\):

\(\displaystyle 7w+6=72\)

\(\displaystyle 7w=66\)

\(\displaystyle {\color{Red} w=\frac{66}{7}}\)

Work problems can be solved by looking at them as rate problems. Therefore, we can look at this problem in terms of tanks per minute, rather than minutes per tank. Let \(\displaystyle X\) be the number of minutes it took to fill the tank.

The pipe filled the tank at a rate of 45 minutes per tank, or \(\displaystyle \frac{1}{45}\) tank per minute; over a period of \(\displaystyle X\) minutes, it filled \(\displaystyle \frac{1}{45} X\) tank.

The drain emptied the tank at a rate of 80 minutes per tank, so we can see this as a drain of \(\displaystyle \frac{1}{80}\) tank per minute. We can look at draining as "filling negative tanks" - \(\displaystyle -\frac{1}{80}\) tank per minute; over a period of \(\displaystyle X\) minutes, it "filled" \(\displaystyle -\frac{1}{80} X\) tank.

Since their work adds up to one tank filled, We can set up, and solve for \(\displaystyle X\) in, the equation:

\(\displaystyle \frac{1}{45} X -\frac{1}{80} X = 1\)

\(\displaystyle \left (\frac{1}{45} -\frac{1}{80} \right )X = 1\)

Using decimal approximations:

\(\displaystyle \left (0.02222 - 0.01250 \right )X = 1\)

\(\displaystyle 0.00972 X = 1\)

\(\displaystyle X = 1 \div 0.00972 \approx 103\) minutes, or 1 hour 43 minutes.

Of the given choices, 1 hour 45 minutes is closest.

In order to solve for x, first subtract 4 from both sides of the equation:

 \(\displaystyle \frac{x}{6} = 6\)

Then, multiply both sides of the equation by 6:

\(\displaystyle x = 36\)

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

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

Solve both equations for y:

\(\displaystyle \newline y = \frac{4+2x}{2} \newline \newline y=-x+8\)

Set them equal to each other and solve for \(\displaystyle x\):

\(\displaystyle \newline -x+8=\frac{4+2x}{2} \newline \newline -x+8=2+x \newline \rightarrow x=3\)

Plug x back into either \(\displaystyle y\)-equation to get

\(\displaystyle y=5\).

To solve for x we want to isolate x on one side of the equation and all other numbers on the other side. To do this we start with adding 5 to both sides.

\(\displaystyle 25=5x-5\)

\(\displaystyle 30=5x\)

Now we divide by 5 to solve for x.

\(\displaystyle \frac{30}{5}=\frac{5x}{5}\)

\(\displaystyle 6=x\)

In order to solve for \(\displaystyle x\), first add 9 to both sides of the equation:

\(\displaystyle 5x = 28 + x\)

Then, subtract \(\displaystyle x\) from both sides of the equation: 

 \(\displaystyle 4x = 28\)

Finally, divide both sides of the equation by 4:

 \(\displaystyle x = 7\)

Add 6 to both sides of the equation:

\(\displaystyle 7x = 21\)

Then divide both sides of the equation by 7:

\(\displaystyle x = 3\)

This problem requires simplification, knowledge of order of operations, and ability to isolate variables and solve for \(\displaystyle x\).

Our first step is to simplify the left side of the equation. Use the distributive property to simplify

\(\displaystyle 3x-2(x-1)\) Distribute the \(\displaystyle -2\)  (Don't forget the negative!!)

\(\displaystyle 3x-2x+2\)  Combine like terms

\(\displaystyle x+2\)

Now we have....

\(\displaystyle x+2 = 4x+3\)

Combine like terms, simplify, and solve for \(\displaystyle x\)

\(\displaystyle x+2 = 4x+3\)    Subtract \(\displaystyle x\) from both sides

\(\displaystyle 2 = 3x + 3\)      Subtract \(\displaystyle 3\) from both sides

\(\displaystyle -1 = 3x\)          Divide by \(\displaystyle 3\) on both sides

\(\displaystyle -1/3 = x\)

In order to solve this equation, we need to move all constants to one side and everything that has an \(\displaystyle x\) to the other side.

\(\displaystyle 4x-\underset{+20}{20}=-9x+\underset{+20}{19}\)

which becomes

\(\displaystyle 4x=-9x+39\)

Moving the \(\displaystyle 9x\) to the left side results in

\(\displaystyle \underset{+9x}{4x}=-\underset{+9x}{9x}+39\)

or

\(\displaystyle 13x = 39\)

Dividing each side by 13 gives us

\(\displaystyle \frac{13x}{13}=\frac{39}{13}=3=x\)