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Algebra Review Help (page 2)

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Updated on Oct 27, 2011

Expressions and Equations

An expression is like a series of words without a verb. Take, for example, 3x + 5 or a – 3.

An equation is a statement that includes the "verb," in this case, an equal sign. To solve an algebraic equation with one variable, find the value of the unknown variable.

Rules for Working with Equations

  1. The equal sign separates an equation into two sides.
  2. Whenever an operation is performed on one side, the same operation must be performed on the other side.
  3. To solve an equation, first move all of the variables to one side and all of the numbers to the other. Then simplify until only one variable (with a coefficient of 1) remains on one side and one number remains on the other side.

Example

7x – 11 = 29 – 3x Move the variables to one side.
7x – 11 + 3x = 29 – 3x + 3x Perform the same operation on both sides.
10x – 11 = 29 Now move the numbers to the other side.
10x – 11 + 11 = 29 + 11 Perform the same operation on both sides.
10x = 40 Divide both sides by the coefficient.
Simplify.
x = 4

Cross Products

You can solve an equation that sets one fraction equal to another by finding cross products of the fractions. Finding cross products allows you to remove the denominators from each side of the equation. Multiply each side by a fraction equal to 1 that has the denominator from the opposite side.  For example...

First multiply one side by and the other by . The fractions and both equal 1, so they don't change the value of either side of the equation.

The denominators are now the same. Now multiply both sides by the denominator and simplify.

ad = bc

This example demonstrates how finding cross products works. In the future, you can skip all the middle steps and just assume that is the same as ad = bc.

Examples

Find cross products.
36x = 6 × 12
36x = 72
x = 2
Find cross products.
4x = 16x + 12
–12x = 12
x = –1

Checking Equations

After you solve an equation, you can check your answer by substituting your value for the variable into the original equation.

Example

We found that the solution for 7x – 11 = 29 – 3x is x = 4. To check that the solution is correct, substitute 4 for x in the equation:

7x – 11 = 29 – 3x

7(4) – 11 = 29 – 3(4)

28 – 11 = 29 – 12

17 = 17

This equation is true, so x = 4 is the correct solution!

Special Tips for Checking Equations on Exams

If time permits, check all equations. For questions that ask you to find the solution to an equation, you can simply substitute each answer choice into the equation and determine which value makes the equation correct.

Be careful to answer the question that is being asked. Sometimes, questions require that you solve for a variable and then perform an operation. For example, a question may ask the value of x – 2. You might find that x = 2 and look for an answer choice of 2. But because the question asks for the value of x – 2, the answer is not 2, but 2 – 2. Thus, the answer is 0.

Equations With More Than One Variable

Some equations have more than one variable. To find the solution of these equations, solve for one variable in terms of the other(s). Follow the same method as when solving single-variable equations, but isolate only one variable.

Example

3x + 6y = 24 To isolate the x variable, move 6y to the other side.
3x + 6y – 6y = 24 – 6y
3x = 24 – 6y
Then divide both sides by 3, the coefficient of x.
x = 8 – 2y Then simplify, solving for x in terms of y.

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