Eli5: I’m struggling to understand linear equations with unknown coefficients

Question

-p (51+z) = dz+84 if I’m solving for Z
First I distribute
Then I move the x terms and the non z terms to opposite sides of the equation.
Thirdly I factor out a z from the z terms
Then I divide both sides of the equation by the factored set of numbers,
and I have my answer.

Z= 84 + 51p/ -p – d

I know how to solve it and get the correct answer 100% of the time, but I don’t know why Im doing any of these steps, or what the answer even means. In comparison to a normal linear equation where there’s just a variable or a number next to a variable(3, 3x) and you solve for a x and you get a number that you can plug into the equation and check to see if you got the right answer.

In: Mathematics

Answer 1

Echoing the other answers, yes solving for z in this case show show it depends on p and d.
Your specific example might just be an example made up for a textbook, but this style of problem comes up in science and math very often. I’ll pull an example from chemistry since it’s what I know best off hand, but keep in mind these types of problems and formulas are everywhere.

Typically you’ll define the ideal relationship between Pressure (P), Volume (V), qty (n, moles), and Temperature (T) of a gas as:
> PV = nRT

If you’re mathematically inclined, this can be straightforward to see the relationship between T and V. So if nothing else changes and my Temperature goes down, then the Volume also goes down, the gas will contract. But what if we move into a higher level chemistry class where we use the van der Waals formula, which captures some non-ideal gas behaviors:
> P = (RT)/(V-b) – a/V^2

Well now it’s less obvious how these values interact. Intuitively, I know that when I cool a gas down, it’s going to try to shrink. But not everything in math and science is so intuitive. Being able to re-arrange this into
> T = (V – b)(a + PV^2 ) / (RV^2 )

Gives you more of an ability to see how T and V relate to one another. This can apply in every science, tech, or math related field. Having the ability to just rearrange a formula correctly is very important.