Of course multiplication and division in algebra are just the same as in arithmetic.
You now know all the arithmetic functions of algebra. Algebra lets you mix and combine these functions.
Yes, there can be more than one equal sign in an equation. Instead of saying,
Side note. You have been solving equations since the first paragraph.
In Algebra How Do You Solve for V? Basic / beginner algebra volume formulas.
About the NASA Formula Examples
Note the "d²" in the volume formula for the cylinder. Yes, the upper "2" means the variable "d" is squared or itself times itself or "d" to the second power.
Note the "a³" in the volume formula for the cube. Likewise, the upper "3" means the variable "a" is cubed or itself times itself times itself or "a" to the third power.
Notice how some of the variables in the formulas are directly adjacent to each other. This is the standard used to indicate the variables are multiplied.
Examples
- The rectangular prism formula or equation, V = a b h, means volume is equal to "a" times "b" times "h".
- The top half of the volume for the sphere formula or equation, "πd³", means pi times d after d has been cubed. If d was equal to 5, then d³ would equal 125, making the equation π times 125 or 125π.
- Yes, the horizontal slash in the sphere and cylinder formulas means divide by the lower number, 6 and 4 respectively.
- As mentioned, "π" is the well-known symbol for pi. The approximate value of pi is 3.14159; this approximation serves most everyday purposes just fine.
More Multiplication Practice
Example #1
A=1
B=2
C=3
D=4
X=A+B*C-D
What is X?
Simplify and solve.
When you see an equation has multiplication and division mixed into it, the rule is to do the multiplication and division first, then do the +’s and -‘s.
So the equation above really means,
X = A + (B*C) - D or
X = 1 + (2*3) - 4 or
X = 1 + (6) - 4
X = 3
The “(“ and “)” are used to indicate what parts of the equation to do first.
It should be noted X=A and A=X are mathematically equivalent.
Just Like the Pros
What you have been and are doing is just simplifying, i.e., breaking down the equation one piece at a time; just like the mathematicians do it. The mathematicians are no more able to look at an equation and instantly come up with the answer any better than the rest of us can. In other words, they can’t grasp the whole equation either. They just solve and proceed from line to line, trusting they solved the previous lines correctly.
Example #2
Here is another one:
A=1, B=2, C=3, D=4, E=5, F=6
((D * B) + (F - 7)) + A) * C = X.
What is X?
This time there is more than one set of parentheses. When that happens, the rule is to do the innermost ones first. So let’s start solving this equation by breaking it down.
The (D*B) and the (F-7) are the innermost parts of the equation.
Let’s start with the (D*B).
D * B = 4 * 2 = 8,
so we simplify the equation to,
(8 + (F-7) + A) * C = X
Next is the (F-7).
F - 7 = 6 - 7
This results in a number one less than zero, so we say negative one or -1.
(Another example would be 15-20. This results in a number 5 less than zero, so we say negative 5 or -5.)
The equation now looks like,
(8 + (-1) + A) * C = X
Let’s get the A and C taken care of; the equation is now,
(8 + (-1) + 1) * 3 = X
Next we add up the numbers inside the parenthesis.
-1 plus 1 equals zero of course.
Or you could have said: -1 plus 8 equals 7. The 8 is called a positive number, just as the -1 is called a negative number. Adding a positive number to a negative number is really just subtracting the negative number from the positive number. In other words:
8 + (-1) = 8 - 1 = 7 or 1 + (-1) = 1 -1 = 0
Either way, our equation now looks like,
(8 - 1 + 1) * 3 = X, which is
(8) * 3 = 24 = X, or
8 * 3 = 24 = X, or
X = 24
Simplified a step at a time and solved.
If you didn’t know negative numbers before, now you do. For the sake of completeness, the next section is about what else one should know about negative numbers.
More About Negative Numbers
Numbers plus negative numbers result in lesser numbers. Keep in mind -10 is a lesser number than -5, etc.
Numbers minus negative numbers result in larger numbers. For example, whereas 9-5 = 4, but 9-(-5) = 14. In other words, minus minus results in a positive increase aka a lesser lesser or a larger larger. Minus a minus is exactly the same as plus a plus, e.g. -(-25)=25.
This is a good time to mention that in mathematics, two negatives equal a positive when applied to minus a minus subtraction, or any multiplication, or any division.
For multiplication:
- Negative numbers times positive numbers equal negative numbers, e.g. -5 * 4 = -20.
- Negative numbers times negative numbers equal positive numbers, e.g. -5 * -4 = 20.
- You already knew positive numbers times positive numbers equal positive numbers.
For division, the same rules apply:
- Negative numbers divided by positive numbers (or vice versa) equal negative numbers, e.g. -5/4 = -1.25 and 5/-4 = -1.25.
- Negative numbers divided by negative numbers equal positive numbers, e.g. -5/-4 = 1.25.
You already knew positive numbers divided by positive numbers equal positive numbers.
More Example NASA Formulas
In Algebra How Do You Solve for V? Learning and doing volume formulas.
Using Spreadsheets
Spreadsheet software or applications will happily do the arithmetic and sort out the negatives versus the positives for you once you have replaced all the variables. It even knows to do the innermost before the outermost, etc. As an example, suppose you have simplified an equation to the following mess:
X=((5-3)* 52)-21+((6+7)/(34-12))
If your spreadsheet software is MS Excel or you are using cloud Google Drive, you can exclude the X and just copy/paste the following into a single cell:
=((5-3)* 52)-21+((6+7)/(34-12))
The spreadsheet will immediately solve the equation and give back the answer of 83.5bunchmoredigits. If you have the software or Google Drive access, go ahead and try it.
If you are experienced at spreadsheet calculations, you can, of course, do equations with the variables still in place; substituting the variables with cell locations or range names.
Another Division Example
Might as well keep it simple and use the previous variables.
A = 5
B = 34
C = 21
X=((A-3)* 52)-C+((6+7)/(B-12))
We replace the variables with the assigned numbers and we are right back where we started from:
X=((5-3)* 52)-21+((6+7)/(34-12))
The arithmetic then gives us:
X = 83.59090909...
Dividing by Zero
This is a good time to mention you cannot divide by zero.
For example:
If A=1
If A=2
If A=3
X = 5 + 10/(3-A)
Now if A=1, then
X=5+10/(3-1)=5+10/2=5+5=10
Now if A=2, then
X=5+10/(3-2)=5+10/1=5+10=15
If, however, we attempt to declare the variable A as A=3, the following occurs:
X=5+10/(3-3)=5+10/0. (invalid)
At this point the equation becomes invalid. There is no answer to the question, “What is 10 divided by 0?”. An equation immediately becomes invalid when a divide-by-zero scenario occurs. Software applications are designed to recognize this when it happens. Plugging whatever-divided-by-zero into a spreadsheet used to give interesting results, before applications were modified to detect this.
What You've Learned
The basic concept of algebra is just plugging the numbers into the variables, and then doing the arithmetic. One merely keeps simplifying the equation until it is solved. You now have a full understanding of that concept. Yes, you have been using variables since the first paragraph.
Final Example
Here is the last example. It is presented in a different format. The question, however, remains the same. What is X? You already know everything needed to solve this equation.
A=1, B=2, C=3, D=4, E=5
T=-1, U=-2, V=-3
(6X/8)+(2T+4)=((CD/2)-AD)+V
It should be noted 6X means the same as 6*X; and AD means the same as A*D. Other examples would be: 3A=3*A=A*3, 5Y=5*Y=Y*5, -2C=-2*C=C*-2, etc.
We plug the numbers into the variables, and the equation now is:
(6X/8)+((2*-1)+4)=((3*4)/2)-(1*4)+-3
Some simplifying arithmetic gives us:
(6X/8)+-2+4=(12/2)-4+-3
More arithmetic then gives us:
6X/8 +2=6-4+-3
More arithmetic gives us:
6X/8+2=-1
We can’t solve X as the equation is currently stated; so we will have to move things around and do more arithmetic.
Important Note
Whenever you change the actual value on one side of the equation, you must do the same on the other side of the equation. Example: 7=7. If you subtract 3 from the left side, then you must subtract 3 from the right side; thus 4=4. The same rule applies for addition, multiplication, and division.
Let’s subtract 2 from both sides of our equation.
6X/8+2=-1
Then becomes:
6X/8=-3
We have to get rid of the “divide by 8” part of the left side of the equation. So we multiply both sides of the equation by 8.
6X/8=-3
Then becomes:
6X=-24
We must make the X stand alone, so we divide both sides by 6.
6X=-24
Then becomes:
X = -4 (The Answer!)
How Do We Know If We Have the Right Answer?
To find out, we go back to the original equation and replace X with -4. We then simplify (reduce) the equation as before to its simplest form. If the simplest possible construct is valid; then, by definition, the statement “X=-4” is valid.
Here is the original equation.
A=1, B=2, C=3, D=4, E=5
T=-1, U=-2, V=-3
(6X/8)+(2T+4)=((CD/2)-AD)+V
We don’t have to re-solve the parts that didn’t have the X in it to begin with, so we have:
(6X/8)+2 = -1
We replace the X with -4, giving us:
((6*-4)/8)+2=-1
Simplifying gives:
(-24/8)+2=-1
Which is:
-3+2=-1
Which is:
-1=-1
This construct is valid and simple enough to know X=-4 is valid.
To take it to the very end, you can multiply both sides by -1, giving us:
1=1
What Would Have Happened, If Instead of Correctly Calculating X=-4, We Had Erroneously Calculated X=16?
The equation simplification/reduction would have proceeded smoothly to this point:
(6X/8)+2 = -1 (as above)
When the 6X is replaced with 6*16, we get:
(96/8)+2=-1 (false)
When further simplified says:
12+2=-1 (false)
Which is
14=-1 (false)
The resulting false statement by definition means the original calculation of "X=16" is a false statement.
The Adventure Continues...
There is a lot more (much, much more) to algebra, but it is really only an expansion of what you have already learned. Algebra is the basis of all other mathematics; including geometry, trigonometry, calculus, and so on. A good understanding of algebra is required to succeed at the other mathematics. Mathematics, itself, is the foundation of most other disciplines. This foundation is not just necessary for the sciences such as physics,
electronics, chemistry, biology, astronomy, and so on. A mathematical foundation is necessary for many careers; including marketing, economics, architecture, and many, many others.
May all your calculations be prosperous ones!
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