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Solving Linear Equations in One Variable
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SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA
SOLVING LINEAR EQUATIONS
 
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SOLVING LINEAR EQUATIONS

Definition:

• A linear equation in one variable can be written in the form

ax + b = c

for real numbers a, b, and c, with a ≠ 0.

Important Properties:
• Addition Property of Equality:
If a, b, and c are real numbers, then

a = b and a + c = b + c

are equivalent equations. (That is, you can add or subtract the same quantity on both sides of the
equation without changing the solution.)

• Multiplication Property of Equality: If a, b, and c are real numbers and c ≠ 0, then

a = b and ac = bc

are equivalent equations. (That is, you can multiply or divide the same nonzero quantity on both
sides of the equation without changing the solution.)

Common Mistakes to Avoid:


• When clearing the parentheses in an expression like 7-(2x-4), remember that the minus sign acts
like a factor of -1. After using the distributive property, the sign of every term in the parentheses
will be changed to give 7 - 2x + 4.

• To clear fractions from an equation, multiply every term on each side by the lowest common de-
nominator. Remember that   is considered one term, whereas, is considered two
terms. To avoid a mistake, clear all parentheses using the distributive property before multiplying
every term by the common denominator.

• To preserve the solution to an equation, remember to perform the same operation on both sides of
the equation.

PROBLEMS

Solve for x in each of the following equations:

No Solution

NOTE: Whenever the variable disappears
and a false statement (such as 72 = 0) re-
sults, the equation has no solution.

All real numbers

NOTE: Whenever the variable disappears
and a true statement (such as 0 = 0) results,
the equation is an identity. An identity is
true regardless of the number substituted
into the variable. As a result, we write \all
real numbers" as our answer.

NOTE: Multiplying each term by the lowest
common denominator of 15 will eliminate all
fractions.

NOTE: Multiplying each term by the lowest
common denominator of 28 will eliminate all
fractions.

NOTE: Multiplying each term by the lowest common denominator of 4 will eliminate all fractions.

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