Summary of Key Concepts
An equation that is true for all acceptable values of the variable is called identity. \(x+3=x+3\) is an identity.
Contradictions are equations that are never true regardless of the value substituted for the variable. \(x+1=x\) is a contradiction.
An equation whose truth is conditional upon the value selected for the variable is called a conditional equation.
Solutions and Solving an Equation
The collection of values that make an equation true are called the solutions of the equation. An equation is said to be solved when all its solutions have been found.
Equations that have precisely the same collection of solutions are called equivalent equations.
An equivalent equation can be obtained from a particular equation by applying the same binary operation to both sides of the equation, that is,
- adding or subtracting the same number to or from both sides of that particular equation.
- multiplying or dividing both sides of that particular equation by the same non-zero number.
A literal equation is an equation that is composed of more than one variable.
Recognizing an Identity
If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.
Recognizing a Contradiction
If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.
Translating from Verbal to Mathematical Expressions
When solving word problems it is absolutely necessary to know how certain words translate into mathematical symbols.
Five-Step Method for Solving Word Problems
- Let \(x\) (or some other letter) represent the unknown quantity.
- Translate the words to mathematics and form an equation. A diagram may be helpful.
- Solve the equation.
- Check the solution by substituting the result into the original statement of the problem.
- Write a conclusion.
A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.
\(>\) Strictly greater than
\(<\) Strictly less than
\(\ge\) Greater than or equal to
\(\leq\) Less than equal to
An inequality of the form
is called a compound inequality.
Solution to an Equation in Two Variables and Ordered Pairs
A pair of values that when substituted into an equation in two variables produces a true statement is called a solution to the equation in two variables. These values are commonly written as an ordered pair. The expression (a, b) is an ordered pair. In an ordered pair, the independent variable is written first and the dependent variable is written second.