# Solving Equations

An equation is a mathematical statement with two expressions that have the same value. The two expressions are separated by an equal sign.

Whatever we do to one side of the equation, we must do the exact same to the other side of the equation to keep the balanced and equal.

## Solving One Step Equations

A one-step equation is an algebraic equation you can solve in just one step. You've solved the equation when you isolate the variable (get the variable by itself on one side of the equation).

This can be achieved by performing the opposite operation occuring on the variable. Remember, operations are simply +, -, *, or /. Addition/Subtration and Multiplication/Division are opposite operations. You must apply the operation to both sides of the equation!

We can solve the following equation for the variable $$x$$. That means, we can solve for the value of $$x$$ that makes this equation true.

$$\cfrac{x}{3} = -8$$

Since the $$x$$ is divded by 3, we need to do the opposite (multiplcation by 3 ) on both sides:

$$\cfrac{x}{3} \cdot \textcolor{red}{3} = -8 \cdot \textcolor{red}{3}$$

On the left side, we have $$x$$ divided by 3 and multiplied by 3. These operations cancel leaving just $$x$$. It is isolated and we can solve for the answer:

$$x = -24$$

Our answer means that this equation is true when $$x = -24$$. You can check your answer by substituting $$x = -24$$ into the original equation:

$$\cfrac{x}{3} = -8$$

$$\cfrac{-24}{3} = -8$$

$$-8 = -8$$

## Solving Two Step Equations

A two-step equation is similar to a one step equation, but invovles one extra step. Most of the time they would take in one of the three forms.

 $$a \cdot x + b = c$$ $$\cfrac{x}{a} + b = c$$ $$a \cdot ( x + b ) = c$$

* The distributive property states that $$a(x+b)$$ equals $$a \cdot x + a \cdot b$$

When you have a two step equation, try the following steps:

1. First, apply distributive property
2. Second, apply addition/subtraction opposite operations
3. Last, apply multiplication/division opposite operations

Solve $$4 \cdot x + 3 = 19$$

Solve $$4 \cdot (x + 8) = 600$$

## Solving Three Step Equations

Similiar to two step equations, but again with the addition of an extra step to solve.

Solve $$\cfrac{2 \cdot x}{5} - 1 = 7$$

## Fractions (Cross Multiply)

Sometimes you may come across $$x$$ in the denominator of a fraction or even fractions on both side of the equation. In these cases, one way to start solving the question is to cross-multiply. The general rule for cross multiply is that if: $$\cfrac{a}{b} = \cfrac{c}{d}$$, then: $$a \cdot d = b \cdot c$$. Don't forget that when you have a whole number, you can make it a fraction by putting a 1 in the denominator.

Find x in $$\cfrac{3}{8} = \cfrac{2}{x}$$