Algebra2 Part 3 Chapter1

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"A 'linear equation' in one variable is an equation that can be simplified to ax + b = c, where 'x' is a 'variable' and 'a', 'b', and 'c' are any 'numbers'. A solution is a value of 'x' that makes the equation true. As long as 'a' is not = 0, this equation has exactly one solution. If 'a'=0, there are two possibilities. If 'b' is not = 'c', no value of 'x' will solve the equation. If 'b' = 'c', then there all real numbers 'x' satisfy the equation; such equations are called 'identities'. Two main tools can be used to solve linear equations: 1. 'The addition property of equality'; Equal quantities may be added to(or subtracted from) both sides of an equation. 2. 'The multiplication property of equality'; Both sides of an equation may be multiplied (or divided) by the same (nonzero) quantity. Note that both of these properties incorporate a general rule of equations you iearned in Algebra 1: do the same thing on both sides of the equation. The addition property is used to move all the variable terms to one side of the equation and the 'constants' to the other side. The multiplication property is often used last to finish getting the variable by itself. Example 1.1 Solve Rx-12 =3x + 7 for x Solution 8x-12 = 3x +7 'Addition (subtraction) property' to move the variables -3x = -3x to the left. _____________ 5x-12 = 7 'Addition property' to move the constants to the right. +12 =+12 ------------------- 5x = 19 Multiplication (division) property to get the variable alone. ---- ---- 5 5 x = 3.8 More complicated equations may require the 'distributive property' to eliminate parentheses. Be careful to multiply signed numbers accurately. Example 1.2 Solve 4x - 2(x + 6) = 3(2x + 7) + 5 for x Solution 4x - 2(x+6) = 3(2x + 7) + 5 4x-2x-12 = 6x+21+5 'Distributive property' to eliminate parentheses. 2x-12 = 6x +26 Combine like terms to simplify the equation. -2x-26 = -2x-26 'Addition (subtraction) property' to get variables on ----------------------- one side and constants on the other. -38/4 =4x /4 ' Multiplication (division) property' to get the variable alone. -9.5 =x or x=-9.5 (A Common Error is to use 2 instead -2)