I’m interested for personal reasons on how these mathematical operations work. I know they’re quite simple, but I never quite understood them, so it’s something that got stuck with me. Any explanations are welcome!
In: 4
Can you please give an example of something you’re questioning?
A linear equation at heart is basically this, take X.
X can be any possible number in the world, any at all. Since X can be an infinity of values, but has no other dimensions, we can draw all the possible values can have on paper, and we call that “a line”.
Of all the infinity of X’s, I can define constraints that “lock” X down to a single, specific number along the X line. That definition usually comes to students in the form of a math equation 2X + 3 -7 = 4. Of all the possible infinite values of X, there is only a single value that makes this equation “true”, since I’m working with a line, and an equation, we call this logic “Linear equations” or “linear algebra”.
What else did you need?
You mean pre-Algebra/Algreba I equations of lines, or solving systems of equations (generally Algebra II)?
A linear equation is just that, an equation for a line.
You the value at an unknown/variable point on the line, x.
You have the value of how drastic or not it changes between positions, slope (m).
You have the starting value (at x=0), the y-intercept (b).
You have the output, y.
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**Example**
y=2x+3
This means that no matter what number x is, you double it and add 3 to get the result/output.
If x = 4, the you have 2*4+3 = 8+3 = 11.
X number line is usually horizontal.
Y number line is usually vertical.
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**Example problem**
3=5x+13
We need to find what number, when multiple by 5 and then adding 13, equals 3. This obviously needs to be a negative value.
5x would need to be -10 (13-10=3)
x then needs to be -2 (-2*5=-10)
Plug it in to check:
3=5(-2)+13
3=-10+13
3=3
It works.
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**Example problem #2**
3x+5 -2x = 6x-10
We don’t really care about the line, mores just x.
So what number if you multiply it by 3, add 5, and then subtract 2 of itself is the same as doing 6 multiples of it minus 10?
On the left had side it can be rearrange:
3x-2x+5=6x-10
Then simplified:
x+5=6x-10
So, if adding 5 to it is the same as multiply by 6 and minus 10, that means x itself is equal to 6 multiples of it self – 15.
x=6x-15
That means 5 multiples instead of 6 would not equal anything.
0=5x-15
This means 5 multiples of x cancels with subtracting 15, so it equals 15.
15=5x
3=x
Get a piece of graph paper. Now draw a small line segment on it, maybe an inch long or so. A little to the left of the line draw a vertical line that starts below your line segment and finishes above it. Now at the bottom of that vertical line draw a horizontal line to the right that finishes past your line segment. You should have something that vaguely looks like the (horrible) diagram below.
Y
l
l /
l
———- X
We can label these things now. The horizontal line we will call the x-axis. The vertical line we will call the y-axis. The point at the bottom where the two lines meet we will call the origin.
Starting at the origin, move to the right on the x-axis until you are below the bottom of the line segment you drew. Now move up from that point until you reach your line segment. The point where you end up can be defined by (X,Y) where X is the number of spaces you moved to the right, and Y is the number of spaces you moved up. You can define any point on your line with two numbers X and Y in the format (X,Y).
Now because of algebra we can use the x and y values in a formula, y=mx+b, that defines your line. The formula is called a linear equation. X and Y are points on the line, m is the slope of your line, and b is where the line crosses the y axis, or the y-intercept. The slope is just how much over and up you need to move to get to the next point on your line. Now we can use that formula to do math things.
edit: Let me work on formatting this.
For equations, let’s take an example:
There is your salary, the taxes, and the money you get at the end. The formula is simply:
* Salary – Taxes = Money
For example, you get paid 10 and taxed 2, you have 8 at the end because
* 10 – 2 = 8
More precisely, if you known that Salary = 10 and the Taxes = 2, then you have
* 10 – 2 = Money
So you can deduce that Money = 8. On the other hand, if you know that the Salary = 20 and Money = 16, then you have
* 20 – Taxes = 16
And if you want to deduce what were the taxes, this is called a linear equation, which mathematicians will write “20-x=16”. To solve this equation, you can either use trial and error and see that Taxes = 4 works, because
* 20 – 4 = 16
Instead of using trial and error, you can use a set of mathematical methods to solve those equations. For example, one method is:
* Starting from “16 = 20-Taxes”, if I undo the “-Taxes” by adding “+Taxes” I obtain “16+Taxes = 20”
* Continuing with “Taxes+16 = 20”, if I undo the “+16” by doing “-16”, I obtain “Taxes = 20-16”, hence “Taxes = 4”.
As you say, a lot of the resolution of equations is knowing “what mathematical operation is undoing which other mathematical operation”. That’s how you get fractions most of the time, you were “undoing a multiplication”, and that’s called creating a fraction.
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