\(\displaystyle 9x + 3x + 2 + 10 = 4(2+x)\)

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

\(\displaystyle 9x + 3x + 2 + 10 = 4\cdot 2+4\cdot x\)

\(\displaystyle 9x + 3x + 2 + 10 = 8+4x\)

Next, we combine like terms. That is, we put all of the terms with an \(\displaystyle x\) on one side of the equation and all the terms without \(\displaystyle x\) on the other. We'll start by subtracting \(\displaystyle 4x\) from both sides:

\(\displaystyle 9x + 3x + 2 + 10 - 4x = 8+4x -4x\)

\(\displaystyle 9x + 3x + 2 + 10 - 4x = 8\)

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

\(\displaystyle 9x + 3x + 2 + 10 - 4x -2 -10 = 8 -2 -10\)

\(\displaystyle 9x + 3x - 4x = 8 -2 -10\)

Combining these terms, this simplifies to:

\(\displaystyle 8x=-4\)

Next, we divide both sides by 8:

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

Which gives:

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

\(\displaystyle \frac{y}{2} + 3 = -7\)

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

\(\displaystyle \frac{y}{2} + 3 -3 = -7-3\)

which simplifies to:

\(\displaystyle \frac{y}{2} = -10\)

From here, we multiply both sides by 2:

\(\displaystyle 2\cdot \frac{y}{2} = -10\cdot 2\)

This gives:

\(\displaystyle y = -20\)

\(\displaystyle 3x+4x+2=5x+6\)

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

\(\displaystyle 7x+2=5x+6\)

Next, we can subtract \(\displaystyle 5x\) from both sides of the equation:

\(\displaystyle 7x+2-5x=5x+6-5x\)

Which simplifies to:

\(\displaystyle 2x+2=6\)

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

\(\displaystyle 2x+2-2=6-2\)

\(\displaystyle 2x=4\)

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

\(\displaystyle \frac{2x}{2}=\frac{4}{2}\)

Which simplifies to:

\(\displaystyle x = 2\)

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

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

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

Which gives:

\(\displaystyle \frac{x}{3}+3x+6=7+x\)

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

\(\displaystyle x+9x+18=21+3x\)

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

\(\displaystyle 10x+18=21+3x\)

Now, we subtract \(\displaystyle 3x\) from both sides:

\(\displaystyle 10x+18-3x=21+3x-3x\)

\(\displaystyle 7x+18=21\)

Next, we subtract 18 from both sides:

\(\displaystyle 7x+18-18=21-18\)

\(\displaystyle 7x=3\)

Last, we divide both sides by 7 to solve for \(\displaystyle x\):

\(\displaystyle \frac{7x}{7}=\frac{3}{7}\)

Which gives,

\(\displaystyle x=\frac{3}{7}\)

To begin, we add 2 on both sides. We get \(\displaystyle 6x = -12\). Dividing by 6 we get\(\displaystyle x = -2\). 

Add the two equations together to get \(\displaystyle 3x=6\). Then divide by \(\displaystyle 3\) to get \(\displaystyle x=2\).

Now substitute this \(\displaystyle x\) value into one of the original equations to get \(\displaystyle 2+3y=-7\). 

Next subtract \(\displaystyle 2\) from both sides to get \(\displaystyle 3y=-9\).

Now divide both sides by \(\displaystyle 3\) to get \(\displaystyle y=-3\).

Therefore the answer is \(\displaystyle (2,-3)\).

Set up a proportion where \(\displaystyle x=\) number of quarts of white paint.

\(\displaystyle \frac{white}{blue} = \frac{x\; qt}{2\; qt}= \frac{2}{1}\) and solve by cross multiplying to get \(\displaystyle x=4\; quarts\), and \(\displaystyle 4 \; quarts = 1\; gallon\).

For percentage problems look for verbal cues:

"IS" means equals.

"OF" means multiplication.

Therefore the equation becomes \(\displaystyle n=1.25(160)=200\).

Since the paintings and the drawings have the same ratio, they will have the same number on display.

Let \(\displaystyle x=\) number of paintings displayed = number of drawings displayed.

Then, \(\displaystyle 140 - 2x =\) number of sculptures displayed.

Set up a proportion:

\(\displaystyle \frac{sculptures}{drawings}=\frac{140 - 2x}{x}=\frac{3}{2}\)

Solve by cross multiplying to get \(\displaystyle 2(140-2x)=3x\) or \(\displaystyle 280 - 4x = 3x\).

Then add \(\displaystyle 4x\) to both sides to get \(\displaystyle 280 = 7x\).

Then divide by \(\displaystyle 7\) to get \(\displaystyle x=40\).

So there are \(\displaystyle 40\) drawings and \(\displaystyle 40\) paintings in the show, leaving the last \(\displaystyle 60\) spots for sculptures.

Verbal cues include "IS" means equal and "OF" means multiplication.

So the equation to solve becomes \(\displaystyle n=1.25\cdot 240=300\)