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Nuances of Pemdas
What is slope intercept form?
It is one of the most recognizable and memorable formulas that all math students / hobbyists recognize. It is normally seen in this general format below
y – y1 = m(x – x1)
Where y1 is the y coordinate, x1 is the x coordinate and m is the slope of that line. The best way to demonstrate this formula is to show how to use it in examples. Let’s start with two separate points, where P1 = (-2,-3) and P2 = (4,5). Let’s find the equation of the line between these two points. The first step in figuring this out is finding the slope of this line first, which is represented with m. The equation for finding the slope of the line is shown below
⦁ How do we find the slope?
Where y2 and x2 are the coordinates within the point P2 and the coordinates x1 and y1 are the coordinates represented by P1. Now we just enter the values into their correct positions based on the formula for the slope, or what I like to call plugging and chugging (probably my favorite phrase of all time in mathematics).
⦁ Plugging and Chugging!
Next, we know that when we subtract negative numbers we it changes the negative number to a positive and we then instead add both of the number instead of subtracting.
After that, we add 5 and 3 together to obtain the value of 8 and then we add 4 and 2 together to obtain the value of 6.
Then we find that value in the numerator (top of the fraction) is 8 and the value in the denominator (bottom of the fraction) is 6. We can simplify this by multiplying both values by to and then cancelling both of the twos
We then get our final answer of the slope to be shown below.
⦁ The slope of our line is:
Now that we know what the slope is, we can now find the equation of the line, remember that the equation of the line is y – y1 = m(x – x1).
⦁ Finding the equation of the our line for points P1 = (-2,-3) and P2 = (4,5)
The first thing that we do is well, plug and chug like when we found our slope! We replace x1 with the x value of P1 and we replace the y1 with the y value of P1. Lastly we replace m with the slope that we also found before. We then get the equation of the form below.
The next thing we do is we change all of our negative values to positive values (since all of our negatives are being subtracted again, they become positive). This step is then shown below.
After we do this, we then multiply 4/3 on the left side to both the x value and the 2 in order to spread out the terms.
We then subtract both sides by three and we isolate y on the left side of the equation and isolate x on the right side of the equation.
Since 8/3 is in fraction form, we need to make sure that we subtract 3 from that value properly so we multiply the top and bottom of the 3 value by three and we get the denominator to be 3 and the numerator to be 9.
Once that has been done we get the final form for the equation of our line at the points P1 = (-2,-3) and P2= (4,5) to be of the form below.
This equation is fine it’s state right now, but what does this equation actually represent? Well, it essentially says that the line has a rise of 4 units and a run of 3 units from left to right. The -1/3 represents the intersection point of the line on the y-axis. We can find this intersection point by setting the value of x = 0 and solving for it like below.
⦁ Finding Intersection Points!
Once we set x = 0, we find that the value of y when x = 0 is actually -1/3. In other words, the point at which the line intersects the y-axis is 1/3 below the x-axis.
Now, let’s say we wanted to find x when the line crossed the x-axis. We just set the value of y to 0 and then solve for x like below.
We have now found that when the line crosses the x-axis, it crosses it when x = ¼ or in other words the line at crosses the x-axis at ¼ unit to the right of the y-axis.