-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
Isolating z can tell you how z depends on those other parameters, for example what happens if p gets very large or very small. And sometimes you have letters like π or g that represent known quantities and you want those terms to remain in your answer.
The order of the steps is so that you systematically peel off or undo the operations that were done to z, so you can eventually isolate z. You start from the outside and work your way in, doing the opposite of what you see. If something was added to z, you subtract it. If something was multiplied by z, you divide by it.
The point of rearranging an equation to get a single unknown in terms of the other terms to to be able to calculate what the value of Z is for any value of p or d.
You could also rearrange the equation to solve for p in terms of z and d or d in terms of z and p.
Btw. Z=(84+51p)/(-p-d). Parentheses are important.
I don’t know the origin of the original equation. But for fun,
Let’s say the original equation describes how far a marshmallow will travel (Z) when shot out of a tube of diameter(d) under air pressure (p).
When solved for Z you can determine how far the marshmallow will go with a 1/2 inch tube and 10 psi of pressure. Or you can rearrange for p, and figure out what pressure you need to hit your big brother 10 feet away.
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.
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