This is from a university placement test practice question. I really don’t know just how to resolve it. There is the answer come the problem and also it verified the process. The answer key said “If 3t − 7 = 5t , climate 5t − 3t −= 7 , and 2t = −7. Erefore, 6t = (3)(2t=) −(3)(−=7) 21“, however I don’t know just how they streamlined it and also doesn’t make any type of sense to me. So can anyone break down on resolving the problem and explain it through extra information please?
There appear to be part typos one of two people in their solution, or in exactly how you typed the out.
You are watching: If 3t − 7 = 5t , then 6t =
3t - 7 = 5t
Put the terms v "t" with each other on the exact same side of the equation. Execute this by subtracting 3t indigenous both sides, and simplifying:
3t - 7 - 3t = 5t - 3t
3t - 3t - 7 = 2t
0 - 7 = 2t
-7 = 2t
Now that we recognize that 2t is, to discover the value of 6t, just multiply the appropriate hand side by 3. However the rules of algebra speak that us must likewise multiply the left hand next by 3. So:
3(-7) = 3(2t)
-21 = 6t
Take the provided equation: 3t-7=5t. You desire to isolate the t ~ above one side of the equation so that you have actually the kind of t=something. In her "toolbox" you can use common math operations +, -, x, / however you have actually to apply it come both sides. For example, if you desire to include 8 to both sides - girlfriend can. That'll it is in 3t-7+8=5t+8
Unfortunately including 8 doesn't assist us in addressing this.
What usually helps is to choose to among the terms that already is in the equation. Let's watch at her equation again 3t-7=5t. We can select to do something v 3t or with -7 or with 5t. Keep in psychic you desire to have actually the terms through t ~ above one side of the equation only. It's somewhat like solving a puzzle.
We already have 5t ~ above one side. Us can shot "moving" it come the other side, yet that would provide us other = 0. Top top the other hand, us can try moving 3t term and it'll offer us miscellaneous = something*t which place the t top top one next only. For this reason let's execute that.
Remember, we can only use +, -, x, / and the hatchet value applied to both sides.
Let's try those operations in order. First one is +: 3t-7+3t = 5t + 3t we can include terms with the exact same t with each other to yield 6t-7=8t... Well that was no help. Following one.
3t-7-3t = 5t-3t Again, adding terms through the same t provides us 0t-7=2t and also we remember the 0*anything is 0 for this reason -7 = 2t! Look us now have t just on one side. Since this is an = sign, we can flip the equation and still be correct, therefore if -7 = 2t climate 2t = -7
So mental a sample - individually a ax of the same value native both sides zeros that on one side and "moves" its negative to the various other side.
What else deserve to we do. Describe our toolbox again +, -, x, /. Us can try this in bespeak too, but due to the fact that I've done plenty of of these over the years I'll aid you the end a bit. Here's a brand-new pattern - separating both sides of one equation through a constant next come a change (like t) will certainly leave the variable through itself, because any type of number / exact same number = 1 so offered 2t = -7 we deserve to divide both sides by 2 and have 2t/2 = -7/2. If girlfriend remember fractions, 2t/2 = 1t = t and also -7/2 = -3.5 so t = -3.5.
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So currently we have the right to "plug" the value of t into 6t and also get 6*(-3.5) which is straightforward to discover on a calculator (or by hand). 6*(-3.5) = -21
What your book did to be a bit different. 6 deserve to be created as 2*3. For this reason we deserve to say 6t = 2*3*t or 6t = (2t)*3 because we know that 2t = -7 ours answer becomes 6t = (2t)*3 = (-7)*3 = -21 This solution requred a bit fewer steps, yet you required to think of factors of 6 instead.