Question

Find the value of y for a given value of x. If y varies directly with x. If y = -252 when x = 84, what is y when x = 74?

Answer 1

Well when they say "varies directly" what they mean to say is that \((y\propto x)\). In other words, \(y=kx\) where \(k\) is a constant!
So from this, we can derive a proportionality ratio:
\[\eqalign{
&y_1=kx_1\phantom{spce}\rightarrow k=\frac{y_1}{x_1}\\
&y_2=kx_2\phantom{spce}\rightarrow k=\frac{y_2}{x_2}\\
&\phantom{y_2=kx_2spcealittle}\downarrow\\
&\phantom{y_2=kx_2spceali}\left[\frac{y_1}{x_1}=\frac{y_2}{x_2}\right] \\
}\]
So now we can plug in point from two co-ordinates \((x_1,y_1)\) and \((x_2,y_2)\). In this case, we are given the points \((84,-252)\) and \((74,y_2)\)
So then we can substitute:
\[\left[\frac{y_1}{x_1}=\frac{y_2}{x_2}\right]\rightarrow\left[\frac{-252}{84}=\frac{y_2}{74}\right]\rightarrow\left[y_2=\frac{-252\times74}{84}\right]\]
And that's the answer!

Answer 2

so I don't have to add or multiply anything up?

Answer 3

@KeithAfasCalcLover

Answer 4

Well I MEAN, you can clean up \(y_2=\frac{-252\times74}{84}\) by multiplying the two top numbers and divide by the bottom number to get an exact answer. Personal Suggestion ;)