What method do I use?!

ABC is an isosceles triangle, these are the lengths of the sides: AB= 4a+3, BC=2b+5, AC= 2a+b The perimeter of the triangle is 32 cm, find the values of a and b. I don't just want the answer I want to know how to get to the answer 🙂

Answers
Hi Kuromi
theo.cushion
22 April 2011
How we can turn that into an equation relating the perimeter with the equations we already have?
theo.cushion
22 April 2011
I have no idea...
kuromi
22 April 2011
AB is one side, along with BC and AC. If we add those together we get the perimeter
theo.cushion
22 April 2011
In other words, 6a+3b+8
kuromi
22 April 2011
$32 = 4a+3+2b+5+2a+b$
theo.cushion
22 April 2011
bingo
theo.cushion
22 April 2011
So it would be 4a+3+2b+5+2a+b?
kuromi
22 April 2011
Can you type out that equation?
theo.cushion
22 April 2011
theo.cushion
22 April 2011
Ah! I get it now! Thank you so much! :)
kuromi
22 April 2011
Which as you say, can simplify to:
theo.cushion
22 April 2011
How do you work out the value of a and b though?
kuromi
22 April 2011
We still have 2 unknowns though, however we know that an isosceles triangle has 2 edges the same side
theo.cushion
22 April 2011
size*
theo.cushion
22 April 2011
OOh yeah.
kuromi
22 April 2011
But, even if we do know the two sides are the same, I don't get how you can find the value of a and b from that? O.o
kuromi
22 April 2011
Is there any more of the question, because there are several possible answers
theo.cushion
22 April 2011
Erm, it is a part b question. The part a question was AB=BC show that 2a-b=1 but I understood that queation.
kuromi
22 April 2011
Okay, that's useful information. We know that AB=BC. So we know what 2 sides are the same length
theo.cushion
22 April 2011
So expand AB=BC into an equation with a's and b's in it.
theo.cushion
22 April 2011
erm, so 4a+3=2b+5?
kuromi
22 April 2011
Exactly!
theo.cushion
22 April 2011
Can you rearrange that to find b in terms of a?
theo.cushion
22 April 2011
almost, we want to keep b on one side of the equation and move all the other terms to the left
theo.cushion
22 April 2011
So 4a-2b=5-3?
kuromi
22 April 2011
would it be 4a-5+3=2b?
kuromi
22 April 2011
yes
theo.cushion
22 April 2011
then we need to simplify that, and turn 2b into 1b
theo.cushion
22 April 2011
or just 'b'
theo.cushion
22 April 2011
Divide the whole equation by 2?
kuromi
22 April 2011
You've got it
theo.cushion
22 April 2011
including the numbers which don't have letters?
kuromi
22 April 2011
so it'll be 2a-1=b
kuromi
22 April 2011
Precisely So becomes so that simplifying gives us and then
theo.cushion
22 April 2011
How can we use that with what we figured out earlier?
theo.cushion
22 April 2011
Would it be 3(2a-1) instead of 3b?
kuromi
22 April 2011
Yes! So try writing it out in full
theo.cushion
22 April 2011
32=6a+6a-1+8
kuromi
22 April 2011
which means 32=12a+7
kuromi
22 April 2011
Not quite, right idea, but be careful multiplying out the brackets on 3(2a-1)
theo.cushion
22 April 2011
Ooops, is it 32=6a+6a-3+8?
kuromi
22 April 2011
Excellent, so to recap: substituting in gives us , expanding the brackets creates
theo.cushion
22 April 2011
can you simplify that further and solve what a is?
theo.cushion
22 April 2011
32=12a+5 12a=17 a = 1.42 (2 d.p)
kuromi
22 April 2011
yes! :D
kuromi
22 April 2011
It can be better to keep it as a fraction until you have solved everything though
theo.cushion
22 April 2011
so how do we figure out b?
theo.cushion
22 April 2011
Do we do 4a+3, so 4x1.42+3 because 2b+5 equals the same as 4a+3
kuromi
22 April 2011
Hang on, missed a tiny mistake earlier on
theo.cushion
22 April 2011
becomes , not
theo.cushion
22 April 2011
oops...
kuromi
22 April 2011
oh.. which was?
kuromi
22 April 2011
we both missed that one!
theo.cushion
22 April 2011
sneaky mistake?
kuromi
22 April 2011
so
theo.cushion
22 April 2011
, substitute in our value for a: , what does b equal?
theo.cushion
22 April 2011
12-5=7 2b=7 b=3.5
kuromi
22 April 2011
How can we check this answer is right?
theo.cushion
22 April 2011
Well done
theo.cushion
22 April 2011
, , ,
theo.cushion
22 April 2011
erm by adding it into the equation 32=6a+3b+8
kuromi
22 April 2011
Exactly!
theo.cushion
22 April 2011
YES! Thank you so much you've helped me a lot!
kuromi
22 April 2011
No problem the things to remember are: Figure out how to turn ALL of the information you have into equations See what you can rearrange, substitute and simplify Always work out a way to verify your answer It never hurts to just play around with the numbers and equations and see where it takes you.
theo.cushion
22 April 2011
and always to check afterwards?
kuromi
22 April 2011
Yes, it's very very easy to make a tiny mistake. Don't be disheartened it's usually easy to fix. That is why showing your workings in exam questions is so important.
theo.cushion
22 April 2011
And you too!
kuromi
22 April 2011
Thank you!
kuromi
22 April 2011
No problem, have a good Easter
theo.cushion
22 April 2011
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