Precalculus: 1.1 Equations in One Variable
Concepts: Solving Linear Equations, Identities, Conditional Equations, Inconsistant Equations, Solving
Rational Equations, Solving Basic Absolute Value Equations
Solving Linear Equations
An equation involves an equal sign and indicates that two expressions have the same value.
x+ 42 = 67(4 −x) is an equation, and means x+ 42 has the same value as 67(4 −x).
Equivalent equations are equations that have exactly the same solution.
Solving an equation typically involves using the rules of algebra to construct a series of equivalent equations until
you determine a numerical solution for an unknown variable. When using algebra, show enough intermediate steps
in your solution to get the correct answer. A good rule of thumb is to write enough so that a classmate could read
your solution and understand all the steps you used without having you explain it to them.
The Addition Property of Equality: If the same number is added to both sides of an equation, the results on
both sides are equal in value (you have constructed an equivalent equation).
x+ 42 = 67(4 −x) is an equation,
x+ 42 + 76 = 67(4 −x) + 76 is an equivalent equation.
There is also a Subtraction Property of Equality, but since you could think of “subtraction” as “adding the negative”,
I am not including it here since it is the same as the Addition Property of Equality.
The Multiplication Property of Equality: If both sides of an equation are multiplied by the same nonzero
number, the results on both sides are equal in value (you have constructed an equivalent equation).
x+ 42 = 67(4 −x) is an equation,
132(x+ 42) = 132 ×67(4 −x) is an equivalent equation.
The Division Property of Equality: If both sides of an equation are divided by the same nonzero number, the
results on both sides are equal in value (you have constructed an equivalent equation).
x+ 42 = 67(4 −x) is an equation,
(x+ 42)
69 =67(4 −x)
69 is an equivalent equation.
Note that you have to be careful with division and multiplication. Make sure you multiply or divide each entire
side of the equation–if you don’t, you will be making an algebra error!
Solving an equation of the form ax +b=cx +d(or even slightly more complicated equations) involves
constructing a series of equivalent equations that ends with the equivalent equation x= a number. The following
steps are required:
1. Clear any parentheses, and simply as much as possible (simplifying is just advice to make things easier).
2. Collect like terms using the Addition Property of Equality if necessary.
3. Isolate the variable term.
4. Use the Division Property of Equality to isolate the variable.
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