First, let's try to connect the question to some concepts that we may be familiar with already. It's important to note that y = mx + by=mx+b, where mm is defined as the \text{slope}slope and bb is defined as the \text{y-intercept}y-intercept. This also happens to be the \text{slope-intercept form}slope-intercept form for the equation of a line! We can use this equation to find our answer.
This question gives you the following information:
\text{1}1. m = -\frac{3}{4}m=−34
\text{2}2. Coordinates (8, 2)(8,2)
So, how can we use this information to find our answer?
Well, if we use (1), we can rewrite our equation y = mx + by=mx+b as the following:
y = -\frac{3}{4}x + by=−34x+b.
Note that all I did was replace the mm variable with the actual number. Well, we still need to find our bb term! We can use (2) to accomplish just that. Input the coordinates (8, 2)(8,2) into our equation. Since x = 8x=8 and y = 2y=2, we have
2=-\frac{3}{4} \cdot 8 + b = -\frac{24}{4} + b2=−34⋅8+b=−244+b
Luckily for us, 4 divides into -24 perfectly times, leaving us with -6
2 = -6 + b2=−6+b
Adding +6+6 to both sides (you try this step and see what happens!), we obtain
8 = b8=b.
Hence, our \text{equation of the line}equation of the line in \text{slope-intercept form}slope-intercept form is y = -\frac{3}{4}x + 8y=−34x+8. We can also multiply each term by 4 to obtain
4y = -3x + 324y=−3x+32 which becomes 3x + 4y = 323x+4y=32 after we add 3x3x to both sides of the equation.
\text{CHECK}CHECK
It's important to note that the equation of the line characterizes the line; that is, every x-value has a unique y-value. We know that when x = 8x=8, we should get y = 2y=2 since (8, 2)(8,2) is a coordinate on the line. Well, let's plug in x = 8x=8 into our solution and, if we get y = 2, we definitely know we're correct.
Using the slope-intercept form and inputting x = 8x=8, we obtain
y = -\frac{3}{4} \cdot8 + 8 = -6 + 8 = 2y=−34⋅8+8=−6+8=2.
So, when x = 8x=8, we get y = 2y=2. This verifies that our solution is correct.
