# Equations

## The Basic Definition

An equation is basically a math statement with an equal sign in the middle, for instance:

• ${\displaystyle 2(4)={\frac {32}{4}}}$

In algebra, equations often include variables:

• ${\displaystyle 7+a=b}$
• ${\displaystyle 6+10=x}$
• ${\displaystyle x=y}$

We are often asked to simplify or solve them.

## Equations: simplifying & solving

They are similar but completely different things.

### Simplifying Equations

Simplifying equations involves cutting out as much as possible so it can make the most sense and give the same message with the least amount of trouble (and typing, for that matter). To simplify an equation, we do as many steps as we are able with the information we have.

This problem can be simplified by combining like terms.

• ${\displaystyle 2a+3a-b+6=c+c+d}$
• ${\displaystyle 5a-b+6=2c+d}$

This problem can be simplified by combining like terms and using the Basic Laws of Algebra.

• ${\displaystyle 8(a-b)+2(a+3b+b)=6a+2(a+4)+b}$

First we do any math we can within the parentheses. In this case, we can combine like terms.

• ${\displaystyle 8(a-b)+2(a+4b)=6a+2(a+4)+b}$

Next we do any multiplication or division, using the distributive property.

• ${\displaystyle 8a-8b+2a+8b=6a+2a+8+b}$

This leaves us with more like terms to combine.

• ${\displaystyle 10a=8a+8+b}$

### Solving Equations

For more detail, see Solving equations.

Solving an equation means finding out the value of a variable.

• ${\displaystyle x=3+7}$

This is a simple one, if you combine like terms by adding 3 + 7, which equals 10, then you get:

• ${\displaystyle x=10}$

You have just solved for x.

Many equations are more complex than this and can't be solved simply by combining like terms. We often must use the properties of equality to isolate the variable so we can get the solution.

• ${\displaystyle x-3=7}$

To isolate x, we must add 3 to both sides of the equation. -3 + 3 = 0, so x will be isolated as the -3 is canceled out.

• ${\displaystyle x-3+3=7+3}$

We combine like terms to find the value of x.

• ${\displaystyle x=10}$

In some problems, we will not be able to determine the value of the variable, but if told to solve for it, we can still isolate it. With the following problem, we can't find the value of x without knowing the value of y, which added to x makes 32.

• ${\displaystyle x+y=32}$

To isolate x, we subtract y from both sides of the equation. y - y = 0, so x will be isolated as the y on the left side is canceled out.

• ${\displaystyle x=32-y}$

Let's try solving for "b" in the more complex simplifying example above. It started out as:

• ${\displaystyle 8(a-b)+2(a+3b+b)=6a+2(a+4)+b}$

We simplified it to:

• ${\displaystyle 10a=8a+8+b}$

We want to isolate b, so first we'll move the 8a out of the right side of the equation.

• ${\displaystyle 10a-8a=8+b}$
• ${\displaystyle 2a=8+b}$

Now we'll remove the 8.

• ${\displaystyle 2a-8=8-8+b}$
• ${\displaystyle 2a-8=b}$

How can we be sure this is right? We can "plug in" a value for either of the variables and solve the other. Since we've isolated for b already, let's pick a value for a. Let's say a stands for 4. We substitute it in the problem.

• ${\displaystyle 2(4)-8=b}$
• ${\displaystyle 8-8=b}$
• ${\displaystyle b=0}$

Once we have a possible solution set, we can check our work by going back to the original problem and plugging in both numbers:

• ${\displaystyle 8(4-0)+2(4+3(0)+0)=6(4)+2(4+4)+0}$
• ${\displaystyle 8(4)+2(4)=6(4)+2(8)}$
• ${\displaystyle 32+8=24+16}$
• ${\displaystyle 40=40}$

This is true. This does not mean, of course, that ${\displaystyle b=0}$ in the problem ${\displaystyle 2a-8=b}$. b could equal 0, but only if a equals 4. Without knowing the value of a, we cannot solve for b further than ${\displaystyle 2a-8=b}$