1.4 Solving Equations

● A variable is a letter which represents an unknown number. Any letter can be used as a variable.

● An algebraic expression contains at least one variable. Examples: a, x+5, 3y – 2z

● A verbal expression uses words to translate algebraic expressions. Example: “The sum of a number

and 3” represents “n+3.”

Review of Terms

● An equation is a sentence that states that two mathematical expressions are equal.

Example: 2x-16=18

To solve an equation means to find the value for the variable that satisfied the equation.

Steps to Solving Equations● Simplify each side of the equation, if needed, by

distributing or combining like terms.● Move variables to one side of the equation by

using the opposite operation of addition or subtraction.

● Isolate the variable by applying the opposite operation to each side.

• First, use the opposite operation of addition or subtraction.

• Second, use the opposite operation of multiplication or division.

● Check your answer.

Examples

● “y” is the variable.● Add 6 to each side to

isolate the variable.● Now divide both sides by

3.● The answer is 5.

● Check the answer by substituting it into the original equation.

3 6 9

6 6

3 15

3 3

5

Check: 3(5)-6=9

15-6=9

y

y

y

Try this. . .

Did you get x = - 4? You were right!

4 8 24x

Equations with No Solution

5( 2) 13 5

5 10 13 5

x x

x x

Some equations have no solution. That means that no value will work in place of the variable. The solution set is the empty set, {}.

5 5x x 10 13

This is a false statement therefore the equation has no solution.

Try this. . .

2 2

3 4x x

Cross multiply to solve.

2( 4) 2( 3)x x Distribute and solve.

2 8 2 6

2 2

x x

x x

8 6 This is a false statement.No Solution. {}

Identity Equations

5( 2) 10 5

5 10 10 5

x x

x x

Some equations have all real numbers as a solution. That means you can plug any number in for the variable and it will work.

5 5x x 10 10

This is a true statement therefore the solution set is all real numbers.

x