combining like terms
The main concept to drive home here is the
order of operations, or what most people might recall as the accronym: PEMDAS... Which stands for
Parenteses, Exponents, Multiplication,
Division, Addition, Subtraction. PEMDAS pretty much tells us the order of priority in which we do the operations as they are listed above. We do anything inside a parenteses first, then we take care of exponents,
then we do multiplication/division depending on which one is the leftmost operation. Some people would argue that multiplication always takes a higher priority, but that is what we have parenteses for. Then last we have addition/subtraction.
We start working on this expression very much like the way we read: from left to right and look for what is going to come up first according to our accronym...
So here we go.
In 7(2x - 4) -(10 - 3x) We see a 2x - 4 in the parenteses we can't combine those since they do not have like terms... same follows for the 10 - 3x in the other parenteses so we move... So what is next. Multiplication. Lets take care of that then.
the 7 is being multiplied by (2x - 4), so we have to distribute. This yields: 14x - 28
we also have -(10 - 3x) that - sign is really a -1 remember this as it will save you from droping a negative sign in the future. In reality the expression we really started with can be seen like this to avoid dropping minus signs 7(2x - 4) + -1(10 - 3x). Anyways we get -10 + 3x when we distribute the negative 1.
This gives us 14x - 28 - 10 + 3x, at which point we add the 14x and the 3x together to get 17x, and the -28 and the -10 to get -38 because they are like terms.
One more time with out the descriptions:
7(2x - 4) - (10 - 3x)
14x -28 + 3x - 10
17x - 38
First, try to see what like terms are in the expression: I notice one variable, x and several constants. When we try to combine like terms, our goal is to simplify the problem so we have the above form re-written as something like ax+b. Now on to finding the coefficients a and b:
First, I would like to distribute the 7 into (2x-4):
7(2x-4) = (7*2x-7-4) = (14x-28). Now we have (14x-28)-(10-3x).
This is where the problem becomes tricky: We have to distribute the -1 implied by the expression into (10-3x). Why do this? Well consider this alone: -(10-3x). This is equal to -1*(10-3x). So distributing, we have:
-1*(10-3x)=(-1*10-(-1)*3x). Whew. Simplify the first term to get -10, and -(-1)*3x = +3x. This yields
Observe that by distributing the -1, we have converted the problem to an "addition" problem:
Now, we can omit the parentheses, as each term is added (or subtracted, same thing):
Lastly, we combine like terms and sum them, to obtain:
Distribute the 7.
Distribute the implied -1.
Combine like terms.