# Chapter 6 - Algebra: Equations and Inequalities - 6.2 Linear Equations in One Variable and Proportions - Exercise Set 6.2 - Page 362: 42

x = 41

#### Work Step by Step

Starting with our equation: 100 = -(x - 1) + 4(x - 6) We start by using the distributive property to distribute the 4 across the second set of parentheses and a -1 across the first set of parentheses. We use -1 because the negative in front of that set of parentheses implies -1 (we typically don't write 1 or -1 in a distributive property or as a coefficient, instead, it is implied). So, we have 100 = (-1)(x) + (-1)(-1) + 4(x) + (4)(-6) This gives us: 100 = -x + 1 + 4x - 24 Then we can regroup the right side of the equation so that our like terms are together. Remember: The sign to the left of a term stay with the term. 100 = -x + 4x + 1 - 24 Next, we combine like terms on the right side of the equation. 100 = 3x - 23 Our variable term is on the right side of the equation. Let's leave it there. We can add 23 to both sides of the equation to get the 3x by itself. 100 + 23 = 3x - 23 + 23 If we complete the arithmetic if adding 23 to both sides, we get: 123 = 3x Now, we can divide both sides by 3. $\frac{123}{3}$x = $\frac{3}{3}$ Completing the arithmetic (division), we get: 41 = x or x = 41 Checking our answer, we can substitute x = 41 back into the original equation: 100 = -(x - 1) + 4(x - 6) 100 = -(41 - 1) + 4(41 - 6) Using order of operations on the right side of the equation, we can work within the parentheses, giving us: 100 = -(40) + 4(35) Now we can multiply on the right side. 100 = -40 + 140 100 = 100 The numbers match when we substitute the solution and simplify the right side of the equation, this tells us that our solution of x = 41 is correct.

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