Literal Equations and Formulas

Question

x+x=s

p   q

Answer 1

This isn't a problem, yet. What was the original form, and what are p and q supposed to represent?

 

All right, that's a proper problem now.

You want an expression for x ("solve for x"). To do this, you must gather all the terms for x together.

But in order to easily pool the factors of x on the left, you need to get them with a common (i.e. identical) denominator.

So, multiply each by an identity using the "other" denominator:

(x/p)*(q/q) + (x/q)*(p/p) = s

Distribute the products:

(qx/pq) + (px/pq) = s

Collect terms and simplify:

((qx + px)/(pq)) = s

(q+p)*x = spq

x = (spq)/(q+p)

And there you are!

Take another look at the strategy above. It's a general strategy for solving algebraic problems:

Goal 1: simplify anything that can be, first

Goal 2: look for ways to combine the coefficients for each power of your target variable (here, you only had your "x" to the first power, so that was easy)

Goal 3: Peel away the layers of operations on your variable, using the reverse operation type: since you had division present, you use a multiplication (of some sort or another) to "clear" it; if it had been an addition, you would have used subtraction, and so forth.

Note that you COULD have treated the (1/p) and (1/q) as multipliers initially, and pooled them directly:

((1/p)+(1/q))*x = s

x = s / ((1/p) + (1/q))

However, that expression for the result, although it is also correct, is perhaps not so easy to punch into a calculator without making an error somewhere.