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

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Discussion Overview

The discussion revolves around rearranging the equation x = tB + (1-t)A to isolate the variable t. Participants explore various algebraic manipulations and strategies to achieve this, with a focus on the steps involved in the rearrangement process.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant requests help to isolate t from the equation, expressing frustration with the presence of two t variables.
  • Another suggests expanding the brackets as a potential first step in the rearrangement.
  • A participant reflects on their long absence from mathematics and expresses uncertainty about handling the equation.
  • Multiple participants discuss moving terms around to isolate t, with suggestions to combine like terms and manipulate the equation accordingly.
  • There is a hint that the equation may not lead to a quadratic form, as no exponents are involved.
  • One participant shares their calculations using specific values for A, B, and t to verify the equation, indicating they are working towards isolating t.
  • Another participant proposes a formula for t based on their manipulations, but others suggest it may not be entirely correct.
  • There are suggestions to factor the equation as a means to isolate t, emphasizing the importance of moving all t-related terms to one side.

Areas of Agreement / Disagreement

Participants generally agree on the need to isolate t but present various methods and steps to achieve this, leading to some disagreement on the correctness of specific approaches and formulas proposed.

Contextual Notes

Some participants express uncertainty about their algebraic manipulations, and there are unresolved steps in the rearrangement process. The discussion reflects a range of approaches without a clear consensus on the final form of the equation for t.

frogtag
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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
 
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Have a hint:
Expand the brackets...

If you are still stuck, show us what you have done so far.
 
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!
 
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...?
 
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?
 
Quick clue ... is this going to be a quadratic?
 
As Treebeard said: DON'T BE HASTY!

Do NOT move over the tB term as fast as you have done..
 
No, it won't be a quadratic... There are no exponents involved
 
frogtag said:
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.
 
  • #10
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...
 
  • #11
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...
 
  • #12
t = (B - A) / (x - A) ?
 
  • #13
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).
 
  • #14
frogtag said:
t = (B - A) / (x - A) ?

You're close, but not quite right.
 
  • #15
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
 

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