Homework Help: 8th Grade Algebra Equations and Solutions

Equations and Solutions for Algebra

Solving Problems with Two Steps

To solve 2-step equations, you'll perform the same inverse operations on both sides of the equation until you can isolate the variable on one side of the equals sign. Inverse operations 'undo' an operation that is part of one side of the equation. For instance, addition is the inverse of subtraction and vice versa. Here's an example where subtraction and division are used as inverse operations to solve for x:

7x + 4 = 32

7x + 4 - 4 = 32 - 4

7x = 28

7x÷ 7 = 28÷ 7

x = 4

In the next example, addition and multiplication are used to solve another 2-step equation. When a number is placed immediately outside of the parentheses, you should multiply it by what's inside of the parentheses.

(x/5) - 3 = 4

(x/5) - 3 + 3 = 4 + 3

(x/5) = 7

(x/5)(5) = 7(5)

x = 35

Distributive Property

You'll also work with multi-step equations where you'll use the distributive property to arrive at a solution. This property states that a(b + c) = ab + ac, and you can use it to help solve equations like this one:

3(x - 4) = 6

3x - 12 = 6

3x - 12 + 12 = 6 + 12

3x = 18

3x÷ 3 = 18÷ 3

x = 6

Combining Like Terms

Terms always contain a number multiplied by at least one variable, like this: 8x. If two terms contain the same variable, and neither term contains any variable that the other does not, they are considered like terms, and they can be combined. For instance, 8x can be combined with 10x or 7x, but it can't be combined with 2xyz or 3y.

To combine like terms means to perform an operation on the two terms, like addition, subtraction, multiplication or division. When you add or subtract like terms, you simply perform the operation on the numbers and leave the variables the same. For instance, 2x + 3x = 5x and 3x - 2x = x (the term '1x' is typically written as just 'x'). Here's an example of an equation that can be solved by combining like terms:

8x + 2x - 3 = 27

10x - 3 = 27

10x - 3 + 3 = 27 + 3

10x = 30

10x÷ 10 = 30÷ 10

x = 3