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Section 2.1 Linear Equations in One Variable

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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.

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Expression or Equation?

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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.

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Steps to Solving a Linear Equation 1.Got fractions? Multiply by the LCD. 2.Got decimals? Multiply by a power of 10. 3.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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Examples

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More Examples

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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

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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.

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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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Still More Examples

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