SECTION A.6 Solving Equations A45 For example, consider the following succession of equivalent equations: + = = = x x x 2 3 13 2 10 5 We conclude that the solution set of the original equation is { }5 . How do we obtain equivalent equations? In general, there are five ways. Procedures That Result in Equivalent Equations • Interchange the two sides of the equation: = = x x Replace 3 by 3 • Simplify the sides of the equation by combining like terms, eliminating parentheses, and so on: ( ) ( ) + + = + + + = + x x x x x Replace by 2 6 2 1 8 3 1 • Add or subtract the same expression on both sides of the equation: ( ) − = − + = + x x Replace by 3 5 4 3 5 5 4 5 • Multiply or divide both sides of the equation by the same nonzero expression: ( ) ( ) − = − ≠ − ⋅ − = − ⋅ − x x x x x x x x x Replace by 3 1 6 1 1 3 1 1 6 1 1 • If one side of the equation is 0 and the other side can be factored, then write it as the product of factors: ( ) − = − = x x x x Replace by 3 0 3 0 2 Need to Review? The Zero-Product Property is discussed in Section A.1, p. A4 CAUTION Squaring both sides of an equation does not necessarily lead to an equivalent equation. For example, = x 3 has one solution, but = x 9 2 has two solutions, −3 and 3. j Whenever it is possible to solve an equation in your head, do so. For example, • The solution of = x2 8 is 4. • The solution of − = x3 15 0 is 5. Some specific types of equations that can be solved algebraically to obtain exact solutions are now introduced, starting with linear equations . Now Work problem 15 1 Solve Linear Equations Linear equations are equations such as + = − = − = x x x 3 12 0 3 4 1 5 0 0.62 0.3 0 DEFINITION Linear Equation in One Variable A linear equation in one variable is equivalent to an equation of the form + = ax b 0 where a and b are real numbers and ≠ a 0.
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