If BC = 6x, CD = 9x + 10, and BD = 16x − 9, what is BC?

Answer 1

Hi Emalia, so it looks like this question is asking about a number line, right? It might look something like this:

<—— B —— C ——— D ——>

where we don't know the distances between the points except for the expressions that are given.

To start off with, we should remind ourselves that the distance between B and D (also known as BD) is equal to the distance between B and C (or, "BC") plus the distance between C and D (or "CD").

Translated into math terms, we can say "BD is equal to BC plus CD", or:

BD = BC + CD

The problem tells you what expressions you can substitute into that equation for each of the quantities. Since BD = 16x - 9 , BC = 6x , and CD = 9x + 10, let's substitute those in, and work on combining like terms and simplifying:

BD = BC + CD

16x-9 = 6x + 9x+10 --> now we can add 6x and 9x to get 15x

16x - 9 = 15x + 10 --> add 9 to both sides to start trying to get x by itself

16x = 15x + 19 --> almost there, subtract 15x from both sides to combine all the x terms together

1x = 19 -- since 1x is the same thing as x, we now know:

x = 19

Ok, so we've gotten rid of that variable! But we're not finished yet -- the question asked us to find "what is BC". So we can use our new knowledge of x = 19 to find an actual value for BC.

All you'll do to finish up is substitute x = 19 into the expression that the problem gave you for BC (do you remember which one it was? You might have to look back and check!) and simplify to find a single number.

If you get stuck anywhere or want to know more about "why" any part of this solution works, let me know and I'll be happy to go into more detail!