Unit 4
Linear Equations and Graphing

An “equation” is a statement of equality, such as x equals y, or x = y. However, equations are not always true statements of equality. For example, 3 + 5 = 8 is a true statement of equality or a true equation, whereas 5 × 9 = 32 is a false statement of equality or a false equation.

If an equation contains a “variable” or unknown number that can take up various “replacement values,” then the equation is called an “open equation.” The equation x + 5 = 5 + x is an open equation with the variable x. It is also a true equation for all replacement values of the variable x. An open equation that is true for all replacement values of the variable is called an “identity.”

If an open equation is true for certain replacement values of the variable and false for others, then the open equation is called a “conditional equation.” The equation 7x − x² = 12 is true when x = 3 or x = 4, but false for all other values of x. Thus 7x − x² = 12 is a conditional equation. If an open equation is false for all replacement values of the variable, then the equation is called, naturally, an “open and false equation.”

An important concern of algebra is to find the solutions of conditional equations—to find those replacement values of the variable that make an open equation a true equation. The replacement values that make the open equation a true equation are called the “solutions” or “roots” of the equation.

An equation of the form ax + b = 0, where a and b are numbers with a ≠ 0, and where x is the unknown or variable, is called a “linear equation in one unknown.” We will start this unit with such linear equations in one unknown. We will then study systems of linear equations in one unknown.

Objectives

After completing this review, you should feel quite comfortable

1. set up a linear equation in one unknown from a problem statement, graph the equation, and solve the equation.
2. determine whether a given system of linear equations is dependent, inconsistent or independent, by examining graphs of the equations composing the system.
3. solve systems of linear equations in two or three unknowns, and explain how to solve systems of linear equations in more than three unknowns.