Algebra Know-It-ALL

Here’s a challenge!

Manipulate the following equation so it contains x all by itself on the left side, and an expression containing
the constants without x on the right side.

x−a+ 5 =−x−b− 7 +c

Solution

This can be done in various ways. They’ll all produce the same result. To avoid making errors with the signs,
let’s change all the subtractions to negative additions before we start rearranging things. That gives us

x+ (−a)+ 5 =−x+ (−b)+ (−7)+c

We can add a to each side, and then simplify the left side. This gets one of the constants out of the left
side, so we have

x+ 5 =−x+ (−b)+ (−7)+c+a

Next, let’s add −5 to each side, and again simplify the left side. This removes another constant from the
left side, so we have

x=−x+ (−b)+ (−7)+c+a+ (−5)

We can add x to each side, and then simplify both sides. That gets rid of the variable on the right side,
leaving only constants there. Now we have

2 x=−b+ (−7)+c+a+ (−5)

Let’s rearrange the right side to get the letter constants in alphabetical order, followed by the numerals.
(That’s not technically necessary, but it will make things more elegant in the end.) That gives us

2 x=a+ (−b)+c+ (−7)+ (−5)

We can divide through by 2, and then add the two plain numbers in the numerator on the right-hand
side, to get

x= [a+ (−b)+c+ (−12)]/2

Constants, Sums, and Differences 195

Table 12-5. Process for solving the equation a−x=b.

Statements Reasons

a−x=b This is the equation we are given

a−x+x=b+x Add x to each side

a=b+x Simplify the left side

b+x=a Transpose the left and right sides

b+x−b=a−b Subtract b from each side

x=a−b Simplify the left side