Chris L. answered • 09/08/22
28 Years of Excellence in Math, English, and ACT & SAT Tutoring
Hi, Peyton.
When we solve for one (or more) variable(s) in an equation, we are tasked with isolating that variable. We seek to determine what other variables, numbers, or constants satisfy, or solve, the equation in terms of that variable.
(A "constant" is another term, essentially, for a number, as numbers don't vary. Variables, on the other hand, do vary: hence their name.)
To solve for a variable, it's often helpful to do some "reverse engineering." What do I mean by this?
We look at the variable to be solved and see what's being "done" to it. How is it being manipulated?
And then we "undo" whatever that is. We do the opposite of what's been done and in the opposite order.
Our equation is this:
-2x + 3y = -9
What is being done to "y" in this equation? And how do we solve for it?
First, we triple y; we multiply it by 3.
We get 3y.
Then we have this other variable "x," which we multiply by -2; we double it and take the negative/opposite of it.
We get -2x.
Then we add -2x to 3y:
-2x + 3y
Note that this could be written 3y - 2x. (Adding a negative is subtracting the positive.)
Then we equate -2x + 3y to -9:
-2x + 3y = -9
Now for the reverse engineering:
In reverse order of what was done to y before, we need to:
1) subtract -2x from both sides of the equation. This undoes the last thing that was done to manipulate y (which was to add -2x to 3y). This is the same as adding 2x to both sides of the equation.
We get:
-2x - (-2x) + 3y = -9 - (-2x) ...which is the same as
-2x + 2x + 3y = -9 + 2x
...which results in
3y = 2x - 9
2) Now we need to undo the next-to-last thing we did to manipulate y (which, in this case, was the first thing we did to y). We multiplied y by 3, so now we divide both sides by 3. This can be thought of, equivalently, as multiplying both sides by 1/3.
We get 3y/3 = (2x - 9)/3
...which results in
y = (2x - 9)/3
Note that the whole quantity, 2x - 9, is divided by 3, not just 9.
With the 1/3 distributed through 2x - 9, we get
y = 2x/3 - 9/3
or...
y = 2x/3 - 3 (...similarly, y = 2/3(x) - 3)
(This last result, our final answer, is read, "two-thirds-x minus three.")
I hope this helps.
Please let me know if you have any questions :)
