SOLVING LINEAR EQUATIONS
Goal: The goal of solving a linear equation is to find the value of the variable that will make the
statement (equation) true.
Method: Perform operations to both sides of the equation in order to isolate the variable.
Addition and Subtraction Properties of
Equality:
Let , , and represent algebraic expressions.
1. Addition property of equality:
If ,
then
2. Subtraction property of equality:
If ,
then
Multiplication and Division Properties of
Equality:
Let , , and represent algebraic expressions.
1. Multiplication property of equality:
If ,
then
2. Division property of equality:
If ,
then
(provided )
Clearing Fractions or Decimals in an Equation:
When solving an equation with fractions or decimals, there is an option of clearing the fractions or
decimals in order to create a simpler equation involving whole numbers.
1. To clear fractions, multiply both sides of the equation (distributing to all terms) by the LCD of all
the fractions.
2. To clear decimals, multiply both sides of the equation (distributing to all terms) by the lowest
power of 10 that will make all decimals whole numbers.
Steps for Solving a Linear Equation in One Variable:
1. Simplify both sides of the equation.
2. Use the addition or subtraction properties of equality to collect the variable terms on one side of
the equation and the constant terms on the other.
3. Use the multiplication or division properties of equality to make the coefficient of the variable
term equal to 1.
4. Check your answer by substituting your solution into the original equation.
Note: If when solving an equation, the variables are eliminated to reveal a true statement such as,
, then the solution is all real numbers. This type of equation is called an identity. On the
other hand, if the variables are eliminated to reveal a false statement such as, , then there is no
solution. This type of equation is called a contradiction. All other linear equations which have only one
solution are called conditional.
Examples:
A.
Check:
(Solution Checks)
B.
Check:
(Solution Checks)
C.
Check:
(Solution Checks)
D.
Check:
(Solution Checks)
E.
(Multiply by the reciprocal)
Check:
(Solution Checks)
F.
Check:
(Solution Checks)
Identity Example:
G.
True Statement
Solution: all real numbers
Contradiction Example:
H.
False Statement
No Solution
I.
Check:
(Solution Checks)
J.
(LCD is 10)
K.
Check:
(Solution Checks)
LINEAR EQUATIONS PRACTICE
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LINEAR EQUATIONS PRACTICE ANSWERS
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25. All real numbers
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27. No Solution
28. All real numbers
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91. No Solution
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97. All real numbers
98.
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100. No Solution
