107 1.1 Linear Equations A linear equation is a first-degree equation because the greatest degree of the variable is 1. 3x + 22 = 0, 3 4 x = 12, 0.51x + 32 = 2x - 6 Linear equations 2x + 2 = 5, 1 x = 8, x2 + 3x + 0.2 = 0 Nonlinear equations EXAMPLE 1 Solving a Linear Equation Solve 312x - 42 = 7 - 1x + 52. SOLUTION 312x - 42 = 7 - 1x + 52 6x - 12 = 7 - x - 5 Distributive property 6x - 12 = 2 - x Combine like terms. 6x - 12 + x = 2 - x + x Add x to each side. 7x - 12 = 2 Combine like terms. 7x - 12 + 12 = 2 + 12 Add 12 to each side. 7x = 14 Combine like terms. 7x 7 = 14 7 Divide each side by 7. x = 2 CHECK 312x - 42 = 7 - 1x + 52 Original equation 312 # 2 - 42≟7 - 12 + 52 Let x = 2. 314 - 42≟7 - 172 Work inside the parentheses. 0 = 0 ✓ True Replacing x with 2 results in a true statement, so 2 is a solution of the given equation. The solution set is 526. S Now Try Exercise 13. Be careful with signs. A check of the solution is recommended. Distribute to all terms within the parentheses. EXAMPLE 2 Solving a Linear Equation with Fractions Solve 2x + 4 3 + 1 2 x = 1 4 x - 7 3 . SOLUTION 2x + 4 3 + 1 2 x = 1 4 x - 7 3 1 2a 2x + 4 3 + 1 2 xb = 12a 1 4 x - 7 3b Multiply by 12, the LCD of the fractions. 12a 2 x + 4 3 b + 12a 1 2 xb = 12a 1 4 xb - 12a 7 3b Distributive property 412 x + 42 + 6 x = 3x - 28 Multiply. 8 x + 16 + 6 x = 3x - 28 Distributive property 14x + 16 = 3x - 28 Combine like terms. 11x = -44 Subtract 3x. Subtract 16. x = -4 Divide each side by 11.
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