|Simplifying with Addition and Subtraction:|
We can use addition and subtraction to get all the terms with variables on one side of an equation, and all the numeric terms on the other. The equations 3x = 17, 21 = y, and z/12 = 24 each have a variable term on one side of the '=' sign, and a number on the other. The equations x + 3 = 12, 21 = 30 - y, and (z + 2) × 4 = 10 do not.
We usually do this after simplifying each side using the distributive rules, eliminating parentheses, and combining like terms. Since addition is associative, it can be helpful to add a negative number to each side instead of subtracting.
For the equation 8 = 20 - z, we can add z to both sides to get
8 + z = 20 - z + z
=> 8 + z = 20. Now subtracting 8 from both sides,
8 + z - 8 = 20 - 8
Therefore, z = 12
Simplifying by Multiplication
When solving for a variable, we want to get a solution like x = 3 or z = 198 etc. When a variable is divided by some number, we can use multiplication on both sides to solve for the variable.
Simplifying by Division
When solving for a variable, we want to get a solution like x = 9 or y = 21 etc. When a variable is multiplied by some number, we can use division on both sides to solve for the variable.
Solve for x in the equation 7x = 280. Since the x on the left side is being multiplied by 7, we can divide both sides by 7 to solve for x:
7x ÷ 7 = 280 ÷ 7
(7x)/7 = 280 ÷ 7
x/1 = 40
x = 40.
Note: Dividing by 7 is the same as multiplying both sides by 1/7.
Directions: Solve the following equations and enter the solution. Also write at least 10 examples of your own.