Solving Algebra Equations with Multiplication and Divisio

This page assumes you know about variables, basic algebraic equations, and how to solve them using addition and subtraction.

In addition to using addition and subtraction to solve equations, we can also use multiplication and division.

Main Rule

The main rule we need to remember is that when we divide or multiply one side of the equation we have to do the same to other side. We also have to make sure that we divide or multiply the ENTIRE side of the equation and not just a portion of it.

Simple Example

We'll take a simple example first:

If 2x = 6, what does x = ?

We can tell by just looking at this that x = 3, however, we can also solve for it. By learning to solve for x, we can then apply this method to more difficult problems where we can't tell the answer just by looking at it.

Solving for x

2x = 6

We want to get x by itself on one side of the equation. We can do this by dividing 2x by 2 or multiplying by ½.

2x (1/2) = 6 (1/2)
(2/2) x = 6/2
x = 3

Let's try a more difficult problem. This time we will need to add and subtract as well.

3x - 6 = 15

It's easiest to do the addition and subtraction first with this kind of equation.

add 6 to both sides
(3x - 6) + 6 = (15) + 6
3x = 21

divide both sides by 3
(3x)1/3 = (21)(1/3)

x = 7

Now we should check our answer by plugging x = 7 back into the original equation:

3x - 6 = 15
3(7) - 6 = 15
21 - 6 = 15
15 = 15

Another Example Problem with 2 Variables

Solve for x in the following equation:

4x + 3y -12 = 24 - y + 2x

Add 12 to both sides

(4x + 3y -12) + 12 = (24 - y + 2x) + 12
(4x + 3y) = (36 - y + 2x)

Subtract 2x from both sides so there is no x on the right side

(4x + 3y) - 2x = (36 - y + 2x) - 2x
(2x + 3y) = (36 - y)

Subtract 3y from both sides so that 2x is alone on one side

(2x + 3y) - 3y = (36 - y) - 3y
(2x) = (36 - 4y)

Divide both sides by 2 so that we get x all alone

(2x)1/2 = (36 - 4y)1/2

x = 18 - 2y

Note that we divided both 36 and 4y by 2 on the right side.

Let's check our answer using the original equation:

4x + 3y -12 = 24 - y + 2x
4(18 - 2y) + 3y -12 = 24 - y + 2(18 - 2y)
72 - 8y + 3y - 12 = 24 - y + 36 - 4y
60 - 5y = 60 - 5y

Things to Remember

• Always perform the same operation to both sides of the equation.
• When you multiply or divide, you have to multiply and divide by the entire side of the equation.
• Try to perform addition and subtraction first to get some multiple of x by itself on one side.
• Always double check you answer by plugging it back into the original equation.

Solving Equations by Addition or Subtraction

Equivalent Equations
An equivalent equation can be obtained from an existing equation in one of four ways.

• Add the same term to both sides of the equation.
• Subtract the same term from both sides.
• Multiply by the same term on both sides.
• Divide by the same term on both sides.

The following four equations are equivalent to x = 5

• Add 3 to both sides: x + 3 = 8
• Subtract 3 from both sides: x – 3 = 2
• Multiply by 3 on both sides: 3x = 15
• Divide by 3 on both sides:

Solving equations

We can use equivalent equations to solve an equation. The solution is obtained when the variable is by itself on one side of the equation. The objective, then, is to use equivalent equations to isolate the variable on one side of the equation.
Consider the equation x + 6 = 14. For it to be considered solved, the x has to be on a side by itself. How can you get rid of the +6 that is also on that side? Remember that a term and its additive inverse add up to 0. The additive inverse of +6 is –6. To write an equivalent equation, subtract 6 from both sides.

Example:

Solve x + 6 = 14

Solution:

x + 6 = 14
x + 6 – 6 = 14 – 6 (Subtract 6 from both sides)
x = 8

Check:
x + 6 = 14
8 + 6 = 14 (Substitute x = 8 into original equation)


To solve the equation y – 4 = 12, we would need to write an equivalent equation with y on a side by itself. To get rid of –4, we would need to add 4 to both sides of the equation.

Example:

Solve y – 4 = 12

Solution:

y – 4 = 12
y – 4 + 4 = 12 + 4 (Add 4 to both sides)
y = 16

Check:

y – 4 = 12
16 – 4 = 12 (Substitute y =16 into original equation)

Writing Algebraic Expressions

Writing Algebraic Expressions

Problem:	Ms. Jensen likes to divide her class into groups of 2. Use mathematical symbols to represent all the students in her class.
Solution:	Let g represent the number of groups in Ms. Jensen's class.
	Then 2 · g, or 2g can represent "g groups of 2 students".

In the problem above, the variable g represents the number of groups in Ms. Jensen's class. A variable is a symbol used to represent a number in an expression or an equation. The value of this number can vary (change). Let's look at an example in which we use a variable.

Example 1:	Write each phrase as a mathematical expression.
	Phrase Expression the sum of nine and eight 9 + 8 the sum of nine and a number x 9 + x

The expression 9 + 8 represents a single number (17). This expression is a numerical expression, (also called an arithmetic expression). The expression 9 + x represents a value that can change. If x is 2, then the expression 9 + x has a value of 11. If x is 6, then the expression has a value of 15. So 9 + x is an algebraic expression. In the next few examples, we will be working solely with algebraic expressions.

Example 2:	Write each phrase as an algebraic expression.
	Phrase Expression nine increased by a number x 9 + x fourteen decreased by a number p 14 - p seven less than a number t t - 7 the product of 9 and a number n 9 · n   or   9n thirty-two divided by a number y 32 ÷ y   or

In Example 2, each algebraic expression consisted of one number, one operation and one variable. Let's look at an example in which the expression consists of more than one number and/or operation.

Example 3:	Write each phrase as an algebraic expression using the variable n.
	Phrase Expression five more than twice a number 2n + 5 the product of a number and 6 6n seven divided by twice a number 7 ÷ 2n   or    three times a number decreased by 11 3n - 11

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Example 4:	A small company has $1000 to distribute to its employees as a bonus. How much money will each employee get?
Solution:	Let e represent the number of employees in the company. The amount of money each employee will get is represented by the following algebraic expression:

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Example 5:	An electrician charges $45 per hour and spends $20 a day on gasoline. Write an algebraic expression to represent his earnings for one day.
Solution:	Let x represent the number of hours the electrician works in one day. The electrician's earnings can be represented by the following algebraic expression:
Solution:	45x - 20

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Summary:

A variable is a symbol used to represent a number in an expression or an equation. The value of this number can change. An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change.

Expressions and variables

An algebraic expression comprises both numbers and variables together with at least one arithmetic operation.
Example:

A variable, as we learned in pre-algebra, is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numerals:

To evaluate an algebraic expression you have to substitute each variable with a number and perform the operations included.
Example:
Evaluate the expression when x=5

First we substitute x with 5

And then we calculate the answer

An expression that represents repeated multiplication of the same factor is called a power e.g.

A power can also be written as

Where 5 is called the base and 3 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.