# Equations and Factoring

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## 6/5 Equations and Factoring

Add  to  and then subtract  from the sum. If  is a positive integer, the result must be an integer multiple of

This problem is a great chance to put the Knowsys method for math to use. Remember, the first step is to read the problem carefully. If you started reading this question and felt a sinking feeling in your stomach when you saw 8x or the words "positive integer" stop and take a moment to reread the problem. It's really not as difficult as it first appears to be. Step 2 of the Knowsys method is to identify the bottom line. In this case we are looking for what the result must be a multiple of.  Next we assess our options. We could write out a formula and try to simplify it or we could pick numbers. In this case it is probably faster to write out a formula but we will work the problem using both methods so that you know how to use either one. Step 4 is to attack the problem. Believe it or not, attitude makes a big difference. Relax, trust in the method, and go step by step. Don't worry if you aren't sure exactly how the problem will work out.
If we write out the formula described in the problem we get the following
$8x+2x-5=$

We can simplify the equation like so

$8x+2x-5=10x-5=5(2x-1)$

If we look at the fully factored form of the equation we can see that we are going to multiply whatever we have in the parenthesis by 5. That means that our result must be a multiple of 5. Now we look at the answers below.

(A)
(B)
(C)
(D)
(E)

You can see that answer (B) matches our prediction exactly. As I mentioned earlier, there is one other way to solve this problem. We can pick numbers for X. The easiest number to pick for X is 1. If we do that the formula becomes

$8x+2x-5=8(1)+2(1)-5=8+2-5=5$

In this case, our formula gives us the number 5. Since we need an answer that must be true, we can eliminate any answers that do not fit and in this case only (B) works (don't forget that step 5 of the Knowsys math method is to loop back and verify that your answer matches the bottom line). Picking numbers can be a great strategy if you get stuck on a problem but it is generally faster to solve the problem by writing out the formula.