Solving Equations with Variables on Each Side...?

Question

I need help with this problem...

2(f-3) + 5 = 3(f-1)

I really do not understand the whole concept and I asked the teacher many times too but nothing seems to work. She even yelled at me...anyways, please explain it thoroughly.

Thanks :o)

Answer 1

OK - f is just a variable. It could be any number. We need to figure out what values of f will make this equation true. First, simplify the 2(f-3) by distributing the 2. That gives us 2f-6, right?

Now the other side, do the same thing. 3(f-1) = 3f-3

... so now we have 2f-6 + 5 = 3f-3

... simplify the left side some more: 2f - 1 = 3f - 3
... now add 1 to both sides: 2f - 1 + 1 = 3f - 3 + 1
... simplify: 2f = 3f - 2
... now subtract 3f from both sides: 2f - 3f = 3f - 3f - 2

-1f = -2

... I'll leave the last step to you.

Answer 2

Sorry to hear your teacher yelled at you!

Here are the steps to solve the equation:

Step 1: If there are parentheses, get rid of them first. In this example, you need to use the distributive property to get rid of the parentheses:
Left side: 2(f - 3) = 2(f) - 2(3) = 2f -6
Right side: 3(f - 1) = 3(f) - 3(1) = 3f - 3
Once you do that, the equation looks like this:
2f - 6 + 5 = 3f -3

Step 2: If there are like terms on either side of the equation, combine 'em. In this example, the -6 and the +5 on the left side are like terms. Combining them (which in this case just means doing the arithmetic) gives you:
2f - 1 = 3f - 3

Step 3: Get all the occurrences of the unknown onto one side of the equation. Usually it "feels" better if the unknown is on the left, but it can be on either side. In this example, to get all the f's to be on the left, you want to subtract 3f from both sides of the equation, because that will get rid of the 3f that's on the right side:
2f - 3f - 1 = 3f - 3f - 3 (In real life, most people would write the -3f's underneath the 2f and the 3f in the equation, but it's hard to make them line up in a text format like this.)
Now simplify the left side: 2f - 3f - 1 is (2-3)f - 1, or -1f - 1.
And simplify the 3f - 3f - 3 = 0 - 3 = -3.
So now the equation says:
-1f - 1 = -3

Step 4: If anything is being added to, or subtracted from, the term that contains the unknown, use subtraction or addition to get rid of it. In our equation, 1 is being subtracted from the term with the f (because it says -1f -1), so we add 1 to both sides to get rid of that:
-1f - 1 + 1 = -3 + 1
-1f + 0 = -2
-1f = -2

Step 5: If the unknown is still multiplied by some number other than 1, divide both sides of the equation by the number. In our equation, the unknown f is multiplied by -1, so we need to divide both sides by -1:
(-1f)/-1 = -2/-1
The beauty of doing that is that the two -1's on the left side cancel, leaving just f. The right side is 2, so we end up with:
f = 2
So 2 is the solution. If you're supposed to use set notation, you can say the solution set is {2}.
In case it helps, here are the same steps, without all the words:
2(f - 3) + 5 = 3(f - 1)
2f - 6 + 5 = 3f - 3
2f - 1 = 3f - 3
2f - 3f - 1 = 3f - 3f - 3
-1f - 1 = -3
-1f - 1 + 1 = -3 + 1
-1f= -2
(-1f)/-1 = -2/-1
f = 2

Answer 3

First you multiply the equations on both sides to simplify the statement.

2(f-3) + 5 = 3(f - 1)
2f - 6 + 5 = 3f - 3

Then you can add the numbers without the 'f's

2f - 1 = 3f - 3

You now want to get the 'f's on the same side. Since 3 - 2 = 1, you would add (-2f) to both sides. You can do this because it keeps the equation equal. If you do something to one side, you must do it to the other.

-2f + 2f - 1 = 3f - 3 - 2f
-1 = f - 3

Now all that is left is to add 3 to both sides to get f on a side by itself.

3 - 1 = f - 3 + 3
2 = f

Hope this helps to understand what is goin on.

Answer 4

Use distributive property:
2f-6+5=3f-3
2f-1=3f-3

Subtract 3f from both sides:
-f-1=-3

Add 1 from both sides:
-f=-2

Divide -1 from both sides:
f=2

Check:
2(2-3)+5=3(2-1)
2(-1)+5=3(1)
-2+5=3
3=3

I hope this helps!

Answer 5

1. distribute

2f-6+5=3f-3

2. simplify

2f-1=3f-3

3. isolate

2f-1=3f-3

a. -2f

-1=f-3

b. +3

2=f

there you go,, good luck,, feel free to email me any other questions if you need help