Algebraic Manipulation: Fundamental
This section explains and goes over how to be able
to manipulate an equation in order to get the answer to it. This will be useful if you are trying to
graph an equation and wish to put it into a proper form, or if you are dealing with systems of equations
(much later on), or if you are simply trying to solve an equation for a single term.
General rule for equations: what you do to one side of the equation, must be done to the other side!!
These will become progressively harder and more complicated, starting simple
and then going into more complex equations and explaining how to deal with the added difficulty.
The point of these is to solve for a single variable, and put it in terms of only that variable. Meaning
if you were solving for X then you would put it in the form of "X=...".
1.) Addition/Subtraction
You have the equation, X+2=5. The only other thing on the side with the X variable is the "+2".
Since it is addition that binds the X with the 2, this means that we have to do subtraction to undo the
"+2". Since we are trying to undo the addition of 2, we must do the subtraction of 2 to
undo it. This would then result in "X=3".
Summary: When there is addition, you must undo it by subtracting the same value that it is being added by!
2.) Multiplication
Now in the last equation, X+2=5, lets says that the X was actually being multiplied by something. This would add another step to it.
So instead of "X+2=5", it is "3X+2=5". In order to solve this, you still must first undo the addition. So once again we subtract 2 from each side,
but now we have 3X=3. For the X to be by itself, we must get rid of the 3. The 3 is being bound to the X by multiplcation, so to undo it, we must
use division. We would need to divide each side by 3. The resultant would now be "X=1".
Summary: When the X is being bound by multiplication with a number, use division to undo it!
3.) Division
Now if we were to change it from "3X" to "X/3", the X is now being bound to the 3 by division. After we isolate the "X/3", we have
"X/3 = 3". Now since the X is being bound by division we must use multiplication to undo the divide by 3. That means we must multiply each side by 3
to undo the X being divided by 3. The result would be "X=9".
Summary: When X is being bound by division with a number, use multiplication to undo the division!
4.) Parentheses
Parentheses are used to set a group of terms aside, and usually they are set aside because all of the terms, as a group, are either being
divided or multiplied by something. Parentheses are usually there because they are bound by some operation, which, as I said, is usually multiplcation
or division. This means that when you do the multiplication or division associated with the parentheses, it must go to
all terms in the parentheses. Usually it is multiplication or division, but sometimes it might be from something special, like a square root,
or a trigonometric function, but trigonometric functions are talked about a little bit in geometry, and mostly in Trigonometry and shouldn't
be talked about in Algebra.
So if we had "2(X+2)=8", we could do one of two things. First, use the rule what we used in the Multiplication section of this page, or we could use
what we talked about here and multiply through the parentheses, to each term, which would undo the parentheses. Both ways are valid but the first one is
simpler. Using what was learned in the Multiplication section, in order to undo the 2 multiplying the parentheses, we must divide by 2. After doing this
to both sides, the 2 multiply the parentheses get divided out by us dividing by 2. The result being:"(X+2)=4". Without anything requiring the parentheses
to be there, the parentheses disappear and we are left with "X+2=4". Afte subtracting the 2, we are left with "X=2".
Another way to have solved the previous problem, "2(X+2)=8", would be to multiply through the parentheses. When you do that you multiply each term
in the parenthesis, the "X" and the "2". Sine the parenthesis was being multiplied by 2, we would end of having"(2*X)+(2*2)=8". Also notice how the 8 was
ignored because it was not in the parenthesis. Then you would have 2X+4=8. Then subtract the 4 resulting in 2X=4. Then divide by 2 and once again we get
"X=2".
Practice. Solve for X in this equation: 2+(2X-8)/4=5
Click for the answer:
X=10
Summary: When there are parentheses, undo what is binding the parentheses together, ie multiplication/division.
5.) Powers
When there are powers, it can complicate things a lot. Sometimes parentheses are held together because everything in the parentheses are being raised to a
power, such as (X+2)2. When you try to get rid of the power, unlike with multiplication or division, you cannot just apply
the power to each term, as in (X+2)2=(X2 + 22). Those two
DO NOT equal each other. This case would involve binomials. The (X+2) is considered a binomial and with it being raised to the 2 power, it is still
treated as a binomial. Doing (X+2)2 will be discussed in the
binomial section.
Now onto the actual powers section. Usually, powers will come into play when X is raised to a power. Such as X2-4=12.
This deals with square roots however. After adding 4 to each side, you will get X2=16. After taking the square root of
each side you get X=4, and X=-4. When dealing with powers it is important to isolate the thing which is being raised to a power. In the previous example
I added the 4 to each side in order to get rid of the -4 on the side with the X2. After isolating the term with the power,
I undid the power by raising each side by the inverse of the power it was being raised to, which was 1/2 in this case (note: when something
is being raised to the 1/2 power it can be called taking the square root of it). If a number is being raised to a power, in order to simplify it
all you have to do is raise it to that power, nothing special needs to happen.
Summary: You must isolate the term with the power in order to eliminate the power, binomials included!
General Tips for solving a problem involving one variable for that variable:
Look for these things when trying to solve the equation:
- Is there only 1 X term? or are there more than one?
- If there is more than one X term, try to get rid of them until there is one X term, usually by moving them to the same side
- If/once there is only one X term, if it is caught up with parentheses, get the parentheses(that is with the X) by itself
- Once the Parenthesis is isolated, undo whatever is causing the parenthesis to be there, ie multiplication or whatever
- Now, without parentheses, isolate the X term and it's done!
To go into more detail for problems(equations) with more than one of the same variable in it:
- If there are no parentheses, simply subtract/add until there is only one X term (unless there are X terms where X is raised to multiple different powers)
- If there are parentheses, try to undo them or by multiply/dividing through
-
Hint, if the parenthesis is bound by division, use multiplication to undo it. If it is bound by multiplication, multiply it through the parenthesis
- After undo'ing the parentheses, subtract/add out the X terms if possible
- Once there is one X term, use normal techniques to solve it
To go to the next page click
next where multiplication involving variables is addressed.
To go back to Introduction to Variables, click
here.