Solve Equation: x = tB + (1-t)A

[frogtag]

Can anyone help me rearrange this equation so that t is isolated please, it's driving me nuts.

x = tB + (1 - t)A

all I can seem to do so far is isolate one of the t's...

1...

x - (1-t)A = tB

2...

x - (1-t)A
---------- = t
B

 

[Kaimyn]

Have a hint:
Expand the brackets...

If you are still stuck, show us what you have done so far.

 

[frogtag]

If it is a homework question its about 16 years late! No it's just been a very long time since my A-levels and having 2 separate t variables is throwing me. But I'll have a go and post my failure in a bit!

 

[frogtag]

Okay ...

x = tB + (1-t)A

x - tB = (1-t)A or x - tB = A(1-t)

x - tB = 1A - tA

x - tB = A - tA

Not much so far...?

 

[Kaimyn]

The key is to get everything with the variable you want onto one side, and everything else on the other.

How can you manipulate that equation to get the t's onto one side? How can you "fuse" the t's together? How can you get rid of the rest?

 

[frogtag]

Quick clue ... is this going to be a quadratic?

 

[arildno]

As Treebeard said: DON'T BE HASTY!

Do NOT move over the tB term as fast as you have done..

 

[Kaimyn]

No, it won't be a quadratic... There are no exponents involved

 

[HallsofIvy]

Okay ...

x = tB + (1-t)A

x - tB = (1-t)A or x - tB = A(1-t)

x - tB = 1A - tA

x - tB = A - tA

Not much so far...?

Add tB to both sides then subtract A from both sides.

 

[frogtag]

I have got that far already, used...

t = 0.5, A = 2.0 & B = 3.0 to prove it,

x = tB + (1-t)A
x = 0.5 x 3.0 + (1 - 0.5) x 2.0
x = 1.5 + (0.5) x 2.0
x = 1.5 + 1.0
x = 2.5

then expanded the brackets,

x = tB + 1A - tA
x = tB + A - tA
x = 0.5 x 3.0 + 2.0 - 0.5 x 2.0
x = 1.5 + 2.0 - 1.0
x = 2.5

then moved bits around,

x - A = tB - tA
2.5 - 2.0 = 0.5 x 3.0 - 0.5 x 2.0
0.5 = 1.5 - 1.0
0.5 = 0.5

maths all seems good, now just working on how to get t on own...

 

[frogtag]

B - A
----- = x - A
t

(B - A) / t = x - A
(3.0-2.0) / 0.5 = 2.5 - 2.0
1.0 / 0.5 = 0.5
0.5 = 0.5

That works...

 

[frogtag]

t = (B - A) / (x - A) ?

 

[Mentallic]

$$x - tB = A - tA$$

look back at this step, and like has already been said, get all the parts with t onto one side, and all the other stuff on the other side. i.e. Add tB and subtract A from both sides (or equivalently, add tA and subtract x from both sides)

Now, use the rule of factoring. The factoring is the opposite to expanding (so, putting the variables back into brackets).

 

[Mentallic]

t = (B - A) / (x - A) ?

You're close, but not quite right.

 

[HallsofIvy]

From x - A = tB - tA you have x- A= (B- A)t.

Since t is multiplied by B- A on the right side, you get t alone by doing the opposite of multiplying