𝑋𝑌𝑍 is a straight line, where
the length 𝑌𝑍 is seven times the length 𝑋𝑌. Given that 𝑋𝑍 equals 120
centimeters, calculate the length of 𝑌𝑍.
Well, the first thing we can do is
actually start to form some equations using the information that we have. So we know that 𝑌𝑍 is seven times
the length 𝑋𝑌. So we can rewrite this as 𝑌𝑍
equals seven 𝑋𝑌. And what I’m gonna do is actually
label this equation one. So that’s our first equation.
And next, what we can do is we can
say that 𝑋𝑍 is equal to 𝑋𝑌 plus 𝑌𝑍 and that’s because we know that 𝑋𝑌𝑍 is a
straight line and 𝑌 is a point on that straight line. So what I’m gonna do is call that
equation equation two.
Okay, so now, what’s my next
step? So the first thing we can do is
actually substitute equation one into equation two. And we can do that because we know
that 𝑌𝑍 is equal to seven 𝑋𝑌 and 𝑌𝑍 also appears in equation two. And what we’re actually doing here
is solving a pair of simultaneous equations using substitution.
So therefore, when we do that, we
get 𝑋𝑍 is equal to 𝑋𝑌 plus seven 𝑋𝑌. And we got that because actually we
substituted in seven 𝑋𝑌 for our 𝑌𝑍. And then when we simplify this, we
get 𝑋𝑍 is equal to eight 𝑋𝑌. And that’s because if we have one
𝑋𝑌 and we have seven 𝑋𝑌s and add them together, we will get eight 𝑋𝑌.
So that’s great because we now have
𝑋 in terms of 𝑋𝑌. So what we do now is actually
substitute in the fact that we know that 𝑋𝑍 is equal to 120. And we get that from the second
part of the question. So when we do that, we get 120 is
equal to eight 𝑋𝑌.
Okay, now, what we can do is
actually solve this to find 𝑋𝑌. So to do that, what we’re gonna do
is actually divide each side of the equation by eight. And that’s because we have eight
𝑋𝑌 and we want single 𝑋𝑌. So if you divide eight 𝑋𝑌 by
eight, we get just 𝑋𝑌, which is what we’re looking for. But remembering that whatever we do
to one side of the equation, we got to do to the other side of the equation. And when we do that, we get 15 is
equal to 𝑋𝑌. So we now know that 𝑋𝑌 is equal
to 15 centimeters.
Okay, great, so what do we need to
do next? Well, next, we need to look at the
question and see well what is it looking for. The question wants us to calculate
the length of 𝑌𝑍. So in order to actually find out
what 𝑌𝑍 is and calculate that, what we’re gonna do is actually substitute our
value of 𝑋𝑌 — so 𝑋𝑌 equals 15 — into equation one. And when we do that, we’re gonna
get 𝑌𝑍 is equal to seven multiplied by 15, which will give us a value of 𝑌𝑍 of
So therefore, we can say that if
𝑋𝑌𝑍 is a straight line, where the length 𝑌𝑍 is seven times the length 𝑋𝑌, and
given that 𝑋𝑍 is equal to 120 centimeters, the length 𝑌𝑍 is 105 centimeters. So we’ve got the answer, great! But what I wanna do here is just a
couple of things, just a sort of check.
First of all, I’m just gonna show
you how we could actually check if that is the correct answer and then give you an
idea of another sort of visual method you could have used. So what we’re gonna do first is
actually look at a little check. And to do that, we’re gonna
substitute 𝑋𝑌 and 𝑌𝑍 into equation two.
So basically, what we know is that
𝑋𝑌 plus 𝑌𝑍 should be equal to 120. So what we’re gonna do, we’ll put
our values in. So we’ve got obviously 105
𝑌𝑍. And we found that 𝑋𝑌 is 15. So we can see that actually yup,
when we add 15 and 105, we will get 120. So we’ve checked it. So it works and that’s equal to our
Okay and the last thing I wanted to
do is actually show you how we could have thought about this question, maybe with
more visual method. Well, if we think that we know that
𝑌𝑍 is seven times the length 𝑋𝑌, then we could have thought of our line, broken
down into these sections. So we have one which is between 𝑋
and 𝑌 and then seven between 𝑌 and 𝑍. So therefore, we would have had one
plus seven, which is eight parts. So we know that actually the full
line 𝑋𝑍 would have eight parts.
So therefore, if we wanted to find
one part, we’d do 120, total length, divided by eight, which is given as 15. So we know that one part will be
worth 15. So therefore, if we wanted to find
𝑌𝑍, well, we know 𝑌𝑍 is gonna be equal to seven parts. So we do seven multiplied by 15
because 15 was the value for one part, which gives us 105, which is the same value
we got using the first method. So this is just another method you
could use if you preferred kinda of visual representation.