Section 2.1 Linear Equations in One Variable

Introduction A linear equation can be written in the form ax = b* where a, b, and c are real numbers and a is not equal to 0. Recall that an equation will have an equal sign, while an expression will not. We solve equations, we simplify expressions.

Expression or Equation?

Solutions The solution to every linear equation is a real number that satisfies the equation. Our strategy for solving is based on the idea of isolating the variable.

Steps to Solving a Linear Equation 1.Got fractions? Multiply by the LCD. 2.Got decimals? Multiply by a power of Apply the distributive property on each side to remove parentheses. 4.Combine like terms on each side. 5.Move variable terms to the left side, non- variable terms to the right side. “Change sides, change signs.” Simplify.* 6.Divide both sides by the coefficient of the variable term.

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Weird Stuff Happens at Step #5 Sometimes, when you’re moving variable terms to the left and non-variable terms the right, all the variables cancel out on both sides of the equation. When this happens, you have two possibilities: 1.Contradiction 2.Identity

Contradictions All the variables cancel out The resulting statement, which is usually 0 = some number, is FALSE The original linear equation has NO SOLUTION, which can be represented symbolically by Ø.

Identities All the variables cancel out The resulting equation, which is usually 0 = 0 or some number = the same number, is TRUE The solution to the original equation is ALL REAL NUMBERS

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