### Video Transcript

If π over seven equals π over
four equals π over 14, which equals six π minus seven π plus two π over three
π₯, find the value of π₯.

This rather strange-looking
equation links the proportion between a number of variables and constants. It tells us that the proportion of
π to ~~π~~ [seven] is the same as the proportion of π to four. These in turn are the same of the
proportion of π to 14 and the proportion in our final fraction.

Now, what this also tells us is
that we can extract any two expressions and set them equal to each other to solve
for π₯. But we do have a problem. Weβre certainly going to be
interested in our final fraction since that contains the variable π₯ that weβre
trying to find the value of. But it also contains three further
variables. Our job will be to make the
numerator in terms of just one of our variables, π, π, or π. It doesnβt matter which we
choose. Weβll get the same result either
way.

Weβre going to in fact choose π,
and then weβre going to choose the first pair of expressions π over seven equals π
over four. By multiplying both sides of this
equation by seven, we get an equation for π purely in terms of π. Weβll then be able to replace π in
the numerator of our final fraction and eventually π as well to give us an
expression just in terms of π. When we times by seven, we get π
equals seven π over four.

Letβs now take the second pair of
expressions: π over four equals π over 14. We need to find a way to replace π
with some expression in terms of π. So weβre going to multiply both
sides of this equation by 14. So π equals 14π over four. Now weβre not going to simplify any
fractions just yet because we might find at the end weβre going to need to add or
subtract fractions. And thatβs easier if their
denominators are equal.

Letβs now consider the numerator of
the expression we are interested in. Itβs six π minus seven π plus two
π. Replacing π with seven π over
four and π with 14π over four, we find that this is equivalent to six times seven
π over four minus seven π plus two times 14π over four. We distribute the six over the
expression seven π over four to give us 42π over four.

Next, we rewrite seven π with a
denominator of four. So itβs 28π over four. Finally, two times 14π over four
is 28π over four. Then we notice that negative 28π
over four plus 28π over four is zero. So the numerator of our fraction is
in fact just 42π over four or equivalently 21π over two.

And so our final fraction can be
expressed as 21π over two over three π₯. In fact, this is equivalent to
multiplying 21π over two by one over three π₯. So itβs the same as 21π over six
π₯.

Now remember, we can set this
fraction equal to any of our other three fractions. And since weβre trying to eliminate
π so we have an equation purely in terms of π₯, letβs set this equal to π over
four. So π over four equals 21π over
six π₯. Now in fact, we know π canβt be
equal to zero since this equation would make no sense. But in fact, we also know that the
coefficient of π on the left-hand side is one-quarter and on the right is 21 over
six π₯. So we can either divide through by
π or equivalently set the coefficients equal to zero. So one-quarter equals 21 over six
π₯. Then we can simplify the right-hand
side to one-quarter is seven over two π₯.

And we might then next take the
reciprocal of both sides so that four equals two π₯ over seven. We didnβt need to perform this
step, but it makes the solving process easier since our variable is on the numerator
of a fraction. To solve for π₯, letβs multiply
through by seven. So 28 equals two π₯. Finally, we need to divide through
by two, and that gives us π₯ equals 14. So given the relation in our
question, we have found that the value of π₯ is 14.