The goal is to isolate the variable on one side.

$\displaystyle \frac{1}{2}x+\frac{3}{4}=9$

Subtract $\displaystyle \frac{3}{4 }$ from each side of the equation:

$\displaystyle \frac{1}{2}x+\frac{3}{4}-\frac{3}{4}=9-\frac{3}{4}$

$\displaystyle \frac{1}{2}x=8\frac{1}{4}$

Multiply both sides by $\displaystyle 2$:

$\displaystyle x=16\frac{1}{2}$

The goal is to isolate the variable to one side.

$\displaystyle \frac{2}{3}x+6\frac{1}{3}=10\frac{1}{3}$

First, convert mixed numbers to improper fractions:

$\displaystyle \frac{2}{3}x+\frac{19}{3}=\frac{31}{3}$

Subtract $\displaystyle \frac{19}{3}$ from both sides:

$\displaystyle \frac{2}{3}x+\frac{19}{3}-\frac{19}{3}=\frac{31}{3}-\frac{19}{3}$

$\displaystyle \frac{2}{3}x=\frac{12}{3}$

Multiply each side by the reciprocal of $\displaystyle \frac{2}{3}$:

$\displaystyle \frac{3}{2}*\frac{2}{3}x=\frac{12}{3}*\frac{3}{2}$

Cross out like terms and multiply:

$\displaystyle \frac{1}{1}x=\frac{6}{1}*\frac{1}{1}$

$\displaystyle x=6$

You are trying to isolate the $\displaystyle \small x$. 

To do this you must first subtract both sides by 2 to get

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

This then becomes a one-step problem where you multiply both sides by 2 to get

$\displaystyle x=4$

Once you've isolated x, it's important to find the lowest common denominator so that you can add the two fractions you're working with. 

Step 1: Isolate x and convert fractions so that they have a common denominator

$\displaystyle x-\frac{1}{4}+\frac{1}{4}=\frac{1}{2}+\frac{1}{4}$

$\displaystyle x=\frac{1}{2}+ \frac{1}{4}=\frac{1\cdot 2}{2\cdot 2} +\frac{1}{4}=\frac{2}{4}+\frac{1}{4}$

Step 2: solve for x

$\displaystyle x=\frac{2}{4}+\frac{1}{4}=\frac{3}{4}, x=\frac{3}{4}$

Cross multiplication is a short-cut that comes from multiplying by the denominators on both sides of an equation. Broken down, it works like this:

$\displaystyle 4\cdot \frac{x}4=\frac{9}x\cdot 4{}{}$

The 4's on the left side of the equation cancel out.

$\displaystyle x=\frac{9\cdot 4}x{}$

Now, do the same with the denominator on the right side.

$\displaystyle x\cdot x=\frac{9\cdot 4}x\cdot x{}$

The $\displaystyle x$'s on the right cancel out.

$\displaystyle x\cdot x=9\cdot 4$

This is simply the result of removing the denominators, then multiplying them on the opposite sides, i.e. cross multiplication.

Now, to finish solving for $\displaystyle x$, simplify both sides.

$\displaystyle x^{2}=36$

Then take the square root to finish.

$\displaystyle x=\sqrt{36}$

$\displaystyle x=\pm 6$

$\displaystyle \frac{2}{3}s+\frac{1}{3}=-\frac{1}{3}$

$\displaystyle \frac{2}{3}s+\frac{1}{3}-\frac{1}{3}=-\frac{1}{3}-\frac{1}{3}$

$\displaystyle \frac{2}{3}s=-\frac{2}{3}$

$\displaystyle \frac{3}{2}\cdot \left(\frac{2}{3}s\right)=\frac{3}{2}\cdot \left(-\frac{2}{3}\right)$

$\displaystyle s=-1$

First, you want to leave all terms with x on one side and all other terms on the other side. To do this, we can subtract 4/3 from both sides. 

We now have 

$\displaystyle \frac{2}{5}x = \frac{2}{3}$

We can now multiply both sides by the reciprocal of 2/5, which is 5/2, to be able to solve for just x.

$\displaystyle x = \frac{2}{3} * \frac{5}{2} = \frac{5}{3}$

Explanation:

The goal is to isolate the variable to one side.

$\displaystyle 1\frac{1}{3}x+7\frac{1}{3}=15\frac{1}{3}$

First, convert the mixed numbers to improper fractions:

$\displaystyle \frac{4}{3}x+\frac{22}{3}=\frac{46}{3}$

Subtract $\displaystyle \frac{22}{3}$from both sides:

$\displaystyle \frac{4}{3}x+\frac{22}{3}-\frac{22}{3}=\frac{46}{3}-\frac{22}{3}$

$\displaystyle \frac{4}{3}x=\frac{24}{3}$

Multiply each side by the reciprocal of $\displaystyle \frac{3}{4}$:

$\displaystyle \frac{3}{4}*\frac{4}{3}x=\frac{24}{3}*\frac{3}{4}$

$\displaystyle x=6$

1.) Add 8 to both sides, removing the "$\displaystyle -8$". It now reads $\displaystyle \frac{1}{2}x=4x+7$

2.) Multiply both sides by 2, removing the $\displaystyle \frac{1}{2}$. It now reads $\displaystyle x=8x+14$

3.) Subtract $\displaystyle 8x$ from both sides, removing the "$\displaystyle 8x$". It now reads $\displaystyle -7x=14$

4.) Divide both sides by "$\displaystyle -7$", resulting in $\displaystyle x=-2$