Balance
The most important thing to remember when working with equations is to always do the same thing to both sides of the equation. This is called balance.
For example, if you add a number on the left side of the equation, then you also must add the same number to the right side of the equation.
x = 3
x + 2 = 3 + 2x + 2 = 5
Keeping things in balance also works for subtraction, multiplication and division. Here is a multiplication example.x + 2 = 3 + 2x + 2 = 5
p = 5
2 • p = 2 • 5
2p = 10
2 • p = 2 • 5
2p = 10
When you take higher level math classes you will learn about more math operations that can be used when solving equations. For all the operations, you need to remember the concept of balance.
Opposites
When solving equations in algebra, doing the opposite operation is usually required. One way to keep your equation balanced is to move things around by doing the opposite because you have to undo operations that have been done to the variable. The opposite of an operation is one way to move things around in an equation.
The opposite of adding three is subtracting three. If you add three to 100 you get 103. If you then subtract three from 103, you’re back where you started. Adding 3 and subtracting 3 are opposites.
The opposite of subtraction is addition.
The opposite of multiplication is division.
The opposite of division is multiplication.
The opposite of multiplication is division.
The opposite of division is multiplication.
Like Terms
In algebra, a term is a part of an expression or equation that is separated from other parts by + or - signs. In the expression 3p + 7 there are two terms: 3p and 7. Here is a 5 term expression:
5a - 72 - 3a - 2xz + 8
In the expression above, notice that there are two terms that have the variable a, two terms with a number without a variable, and one term that has the variables x and z together.
The 5a and the -3a terms are called "like terms" because they both have the variable a. They are alike!
The number -72 and the number +8 are also like terms. The are alike because they are both pure numbers with no variable next to them.
The -2xy term is different from all the other terms and cannot be combined with anything.
The 5a and the -3a terms are called "like terms" because they both have the variable a. They are alike!
The number -72 and the number +8 are also like terms. The are alike because they are both pure numbers with no variable next to them.
The -2xy term is different from all the other terms and cannot be combined with anything.
When you have like terms in algebra, they can be combined. If you put together the 5a and the -3a you get 2a because 5a - 3a = 2a.
You can also put together the - 72 and the +8 terms and get -64 because -72 + 8 = -64
You can also put together the - 72 and the +8 terms and get -64 because -72 + 8 = -64
Once you put together all the like terms, your expression is usually much simpler:
5a - 72 - 3a - 2xz + 8
2a - 64 - 2xz
2a - 64 - 2xz
Parentheses
In most equations (except those with fractions) you need to get rid of parentheses before you can do anything else. The way to do this is called the Distributive Property.
You know that in algebra when two things are next to each other it means that the two things are multiplied. For example 3e means 3 times e. This is also true when you have something next to a parentheses: 2(e + 3) means 2 times e plus 2 times 3. In algebra you would write it this way:
2(e + 3) = (2 • e) + (2 • 3) = 2e + 6
For most students, you can do the multiply step in the middle in your head, so your solution would look like this:
2(e + 3) = 2e + 6
You can also use the Distributive Property with subtraction:
5(e - 7) = 5e - 35
The general rules for the Distributive Property is given below. The letters a, b, and c stand for any number or variable you want to use.
Operation | Example |
| Addition | |
| a (b + c) = ab + ac | 4(7 + a) = 28 + 4a |
| (b + c)a = ba + ca = ab + ac | (w + 6)3 = 3w + 18 |
| Subtraction | |
| a (b - c) = ab - ac | 8(f - 2) = 8f - 16 |
| (b - c)a = ba - ca = ab - ac | (9 - b)2 = 18 - 2b |
Fractions without Parentheses
Fractions are probably one of the hardest things to do in arithmetic. Unfortunately, sometimes you need to solve an algebra equation that has fractions. The easiest way to do this is to get rid of the fraction, something your arithmetic teacher would never let you do!
If you slept through fractions in elementary school, here are terms you need to know:
- numerator - the top part of a fraction
- denominator - the bottom part of a fraction
- mixed number - a number like 5½ that has both an integer part (5) and a fraction part (½). Algebra people do not like mixed numbers. Turn it into an improper fraction as quickly as possible.
- improper fraction - a fraction where the top is bigger than the bottom. The mixed number 5½ can be turned into an improper fraction by multiplying the integer part by the denominator and then adding the numerator (5 • 2) + 1 so the improper fraction is 11/2
To get rid of fractions, first figure out the common denominator. The easiest way is to multiply all the denominators together - this might give you a bigger number than you really need, but it works if you are desperate.
Now multiply everything on both sides of the equation by the common denominator. The new equation will not have any fractions - hurray! Here is an example:
¼x + 3 + 2x = 5 | 4 is the common denominator | ||||
| (4 • ¼x) + (4 • 3) + (4 • 2x) = 4 • 5 | multiply everything by 4 | ||||
| x + 12 + 8x = 20 | simplify your answer | ||||
| 9x + 12 = 20 | combine like terms - now you have a two-step equation | ||||
| 9x + 12 = 20 -12 -12 | subtract 12 from both sides of the equation | ||||
9x | = | 8 | divide both sides by 9 | ||
| 9 | 9 | ||||
| x | = | 8 | you are done! your answer has a fraction, but it wasn't hard to do | ||
| 9 | |||||
Fractions with Parentheses
When you have a fraction and parentheses the rules change a little bit. You still need the common denominator. If you forgot how, check the section on fractions without parentheses. Once the fraction is gone, then you will use the Distributive Property to help you get rid of any parentheses that are left.
This time we will only multiply the stuff in parentheses and everything on the other side of the equation by the common denominator. Here is an example:
½(x + 5) = 6 2 • ½(x + 5) = 2 • 6 x + 5 = 12 | 2 is the common denominator multiply both sides by 2 simplify your answer, a one-step equation |
Steps for solving equations
Always try to solve equations in the order below. It might work if you use a different order, but it will add steps and make it harder. Most algebra problems only have some of the steps - not all those listed below.
- First - get rid of any fractions
- Second - get rid of any parentheses
- Third - combine like terms to get all the variable terms together and the number terms together
- Fourth - move all the variable terms by adding or subtracting (using the property of opposites) on both sides of the equal sign so the variable terms are all on one side of the equal sign
- Fifth - move any number terms by adding or subtracting (using the property of opposites) on both sides of the equal sign so the number terms are all on the other side of the equal sign from the variable term
- Sixth - if there is a number multiplied by the variable, get rid of it by dividing
Examples
| 3y = 15 | Original Problem |
| 3y ÷ 3 = 15 ÷ 3 | to get rid of the 3 multiplied by the y, divide both sides of the equation by 3 |
| y = 5 | this is your answer |
| 2w - 4 = 8 | Original Problem |
| 2w - 4 + 4 = 8 + 4 | move the -4 away from the 2w term by adding 4 to both sides of the equation |
| 2w = 12 | combine like terms, -4 + 4 = 0 and 8 + 4 = 12 |
| 2w ÷ 2 = 12 ÷ 2 | to get rid of the 2 multiplied by the w, divide both sides of the equation by 2 |
| w = 6 | this is your answer |
Example with fractions and no parentheses
| ¾r - 2 = 1 | Original Problem |
| (4 • ¾r) - (4 • 2) = 4 • 1 | get rid of the fraction by multiplying everything by the common denominator 4 |
| 3r - 8 = 4 | do the math |
| 3r - 8 + 8 = 4 + 8 | move the -8 away from the 3r term by adding 8 to both sides of the equation |
| 3r = 12 | do the math |
| 3r ÷ 3 = 12 ÷ 3 | to get rid of the 3 multiplied by the r, divide both sides of the equation by 3 |
| r = 4 | this is your answer |
Example with fractions and parentheses
| ¼(g + 2) = 8 | Original Problem |
| 4 • ¼(g + 2) = 4 • 8 | get rid of the fraction by multiplying both sides by the common denominator 4 |
| g + 2 = 32 | do the math |
| g + 2 - 2 = 32 - 2 | move the +2 away from the g term by subtracting 2 to both sides of the equation |
| g = 30 | this is your answer |
No Solution and Identity
You can't always get just one answer in algebra.
Sometimes when you are trying to solve a simple algebra problem you get a really weird looking answer. If the answer is a big fat lie like 5 = 3 or -7 = 2, we know that there is no answer to the problem. The algebra way to say big fat lie is no solution. This impresses people and they think you are doing some really hard stuff. As soon as you notice you have a big fat lie you get to stop and write down no solution for your answer. Here is an example:
| 2 (x + 4) = 3x - x + 5 | Original Problem |
| 2x + 8 = 3x - x + 5 | distribute 2 times x + 4 |
| 2x + 8 = 2x + 5 | combine like terms 3x - x |
| 2x - 2x + 8 = 2x - 2x + 5 | subtract 2x from both sides of the equation |
| 8 = 5 | combine the like terms 2x - 2x = 0 |
| No solution | this is your answer because 8 does not equal 5 ever! |
Another weird answer that sometimes happens is something that is always true like a = a or 6 = 6. This means that you can pick any answer at all to put back into the original equation and you will always get a true statement. The algebra way to say always true is identity. As soon as you notice you have something that is always true you get to stop and write down identity for your answer. Identity is another way to say all real numbers!.
Here is an example:
| 2 (a - 4) = 5a - 3a - 8 | Original Problem |
| 2a - 8 = 5a - 3a - 8 | distribute 2 times a - 4 |
| 2a - 8 = 2a - 8 | you get to stop now because you notice that you have the same stuff on both sides of your equal sign |
| Identity or all real numbers | this is your answer because 2a - 8 always equals itself! |
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