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Thinking Mathematically Algebra: Equations and Inequalities 6.2 Solving Linear Equations “Solving” Linear Equations Two algebraic expressions connected with an “equal” sign is called an “equation.” When there are no exponents for the variables (other than one) the equation is called “linear.” “Solving” an equation means finding all of the numbers for the variables that make the equal sign true. For equations with only one variable, the “solution” is the set of all of those numbers. The Addition Property of Equality The same real number (or algebraic expression) may be added to both sides of an equation without changing the solution set. This can be expressed symbolically as follows: If a = b, then a + c = b + c. The Subtraction Property of Equality The same real number (or algebraic expression) may be subtracted from both sides of an equation without changing the solution set. If a = b, then a – c = b – c. “Solving”Linear Equations The rules of equality can be used to “solve” a linear equation. Any number or variable can be added or subtracted from both sides. The “goal” is to isolate the variable on one side of the equal sign. Exercise Set 6.2 #3 x + 5 = -12 The Multiplication Property of Equality The same nonzero real number (or algebraic expression) may multiply both sides of an equation without changing the solution set. If a = b and c ≠ 0, then ac = bc. The Division Property of Equality Both sides of an equation may be divided by the same nonzero real number (or algebraic expression) without changing the solutions set. If a = b and c ≠ 0, then a/c = b/c. “Solving”Linear Equations The rules of equality can be used to “solve” a linear equation. Both sides of an equation may be multiplied or divided by any (non-zero) number or variable. The “goal” is to isolate the variable on one side of the equal sign. Exercise Set 6.2 #7 5x = 45 Solving a Linear Equation 1. Simplify the algebraic expression on each side. 2. Collect all the variable terms on one side and all the constant terms on the other side. 3. Isolate the variable and solve. 4. Check the proposed solution in the original equation. Exercise Set 6.2 #19, #23, #41 14 – 5x = -41 5x – (2x – 10) = 35 6 = -4(1 – x) + 3(x + 1) Misc. • Solving linear equation with fractions Exercise Set 6.2 #47 x x 5 3 2 6 •Special cases: no solution, all real numbers Exercise Set 6.2 #75, 81 2(x + 4) = 4x + 5 – 2x + 3 4(x + 2) + 1 = 7x – 3(x – 2) Thinking Mathematically Algebra: Equations and Inequalities 6.2 Solving Linear Equations