Solving Systems, But Better
Using a graph for salving more complex systems kinda sucks, so you’ll need to find another way to solve those juicy systems.
Well let my introduce the best way to solve systems, elimination!
So when you have a system, you can add them together. when you add them together (make sure the x or y can equal 0)(also can multiply/divide an equation to get it good) it’ll make the system into 1 single equation, where if you find the value for x or y (in the example, is x), you can also find the other missing value by imputing the newly discovered value into one of the system’s equations.
In the example above, there is a lot of tedious math (mainly because I accidentally made it this way) and make sure you verify your answer to ensure that the answer is correct.
A system is a pair of linear equations that have the same x and y values (a point in common on a graph)
You can try to figure it out in your head, or you can graph the equations and use sick visuals to find the solution.
In this graph, we have a system for lines x + 4 = y and 4x + 10 = y. When you take a look at the lines, they seem to intersect at a specific point. This point is (-2,2) and is the solution to the equation
In the graph, we have some really ugly points and for this equation, using a graph is helpful for understanding systems, but don’t rely on it because it takes too long and sometimes it not very accurate.
Point Slope Form
I would say that point slope is my favorite equation for a line. It tells you where a nice point is, what the slope is, and it looks best in my opinion.
To graph this type of equation you’ll need to start at the nice point, and use the slope to create the line.
What Is Slope?
The slope of a line represents how steep the line is on the graph. To find the slope, you’ll need to identify the nice points (Whole numbers) on the line (at least 2).
Now you need to find the rise and run that is required to get from 1 point to the next (rise before run). After you know your rise and run, you can put it in a fraction (rise over run). This fraction will be your slope (simplify if possible). Also we use the letter m for slope.
In this example we have a rise of 5 and a run of 1. So our slope will be 5 (m = 5).
Make sure to read left to right, because sometimes you’ll come across a negative slope. The difference between positive and negative slopes is that a positive slope is going up and a negative slope is going down (reading from left to right)
In this example, the rise is -1 and the run is 1, so the slope will be -1 (m = -1)
How to Find the Distance Between 2 Points on a Graph
We have 2 points ( (2,4) and (6,4), to find the distance between these 2 lines we’ll need to see if any of the x or y values are the same. If they are, it makes your job in finding the distance easier. In this example, our y values are the same, so all we have to do is subtract 2 from 6 (6 – 2) and well get 4. So the distance between these 2 points will be 4.
If the line is not perfectly vertical or horizontal, Finding the distance will take a bit longer
in this graph, we have points at (1,1.5) and (2,4). If you look closely, you’ll see a right triangle like this
So all we have to do is some Pythagorean theorem and we can find our answer.
Functions Will have names (it mostly matters when you have more than one) and typically people will use F, but it can be any letter.
The relation will tell us what to do when we have certain values
So if we have the function name and a 4 instead of and x, we will replace all the x values with 4. and at that point we would be able to find what the Y value would be if the x value was 4.
You can also get something where they tell you the Y value and you need to find the x value. To do this you’ll need to reverse the steps from the first example. So here we will subtract 21 by 7, then divide it by 2 and then there is our answer.
Domain And Range
Domain and range are math terms that describe what possible numbers that can be the x and y values in a graph (domain = x, Range = y)
There are two graphs in this photo. The one on top is a discrete graph and for the domain and range, you will list off all the x and y values that are used for each dot. The graph below has a line that starts at (0,2). (note: lines are just a bunch of dots but they are connected) So for the domain, you will state that 0 is smaller than or equal to x. For the range, you will state that 1 is smaller or equal to y.
Here is some other examples of graphs. the one on the left has a domain of “All Real Numbers” because there is a slight angle to the right and left and it doesn’t have an ending dot. For the range, you will you will state that -2 is smaller or equal to y (remember, lines are just a bunch of dots connected together). The graph on the left has an open dot, this says that x or y wont be included for that specific dot. So the domain will state that 1 is only smaller than x and that x is smaller or equal to 6. The range can only be 1 because the line stays on 1 for the entire time.
Here is one last example of a graph with wavy lines (Remember, Lines are a bunch of dots that are connected). so the domain will state that -5 is smaller than or equal to x and that x is smaller or equal to 3, and the range will state that -2 is smaller or equal to y and y is smaller or equal to 5.