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1.1 Linear Equations A linear equation in one variable is equivalent to an equation of the form To solve an equation means to find all the solutions of the equation. ax b 0 where a and b are real numbers and a 0. Equivalent Equations Two or more Examples of equations that have equivalent precisely the same equations: solutions are called 3 x and x 3 equivalent ( x 2) 6 2 x ( x 1) and equations. x 8 3x 1 3x 5 4 and (3x 5) 5 4 5 Solving a Linear Equation Solve the equation: 3x 5 4 (3 x 5) 5 4 5 3x 9 3x 9 3 3 x3 We replace the original equation by succession of equivalent equations. Add 5 to both sides. Simplify. Divide both sides by 3. Simplify. Solving a Linear Equation Involving Fractions Solve the equation: x 1 x 5 2 6 x x 1 30 30 6 5 2 x 1 x 30 30 30 5 2 6 6 x 15 5 x 6 x 5 x 15 5 x 5 x x 15 0 x 15 15 0 15 x 15 Multiply both sides by the least common denominator 30. Be sure to multiply all terms by 30. Divide out common factors. Subtract 5x to get the x-terms on the left. Simplify. Solve an Equation Involving Rational Expressions Solve: 4 9 7x 4 x 5 5x 4 9 7x 4 x 5 5x 4 9 7x 4 5x 5x x 5x 5 4 9 7x 4 5x 5x 5x x 5 5x 20 9 x (7 x 4) 20 9 x 7 x 4 20 2 x 4 20 4 2 x 4 4 16 2 x 16 2 x 2 2 8 x This is the given equation. Multiply both sides by 5x (LCD). Be sure to multiply all terms by 5x. Divide out the common factors. Simplify. Subtract 4 from both sides. Divide both sides by 2.