Question

WHOEVER EXPLAINS ALL THEIR WORK AND GOES THROW EAXH STEP WILL GET BRAINLEST THAT IS COMPLETELY FINISEHD. write the equation of the line that goes through the points (3,2) (4,-1) show ALL steps and EXPLAIN what you are doing in each step. you will be graded on showing all the steps and your explanation.

Answer 1

Answer:

$y = -3x+11$

Step-by-step explanation:

I accept your challenge!

First we know that a equation of the line is a linear equation, and its equation is:

$y = mx+b$

$m: \text{slope}\\b: \text{y-intercept}$

In this context, the slope, defined as the steepness of a line is characterized as the ratio of RISE (the difference in the y-coordinates, the vertical change) over the RUN (the difference in x-coordinates, the horizontal change).

$m=\frac{\text{change in y}}{\text{change in x}}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Once we know two points where the line goes through, we can get the slope, once we know the change in y and x.

I will take $(3,2)$ as Point 1 and $(4,-1)$ as Point 2.

So, $x_{1}=3 \text{ and } y_{1}=2\\x_{2}=4 \text{ and } y_{2}=-1$

$m=\frac{\text{change in y}}{\text{change in x}}=\frac{\Delta y}{\Delta x}=\frac{-1-2}{4-3}$

$m=\frac{-3}{1}=-3$

Now we have to find the y-intercept:

To do so, think about the the points given and the nearly finished equation, we can find b with those two informations.

Our currently equation of the line is:

$y = -3x+b$

Using the points, take Point 1 as an example, it says that when $x = 3, y = 2$, so plugging it in the equation:

$2 = -3(3)+b\\2 = -9+b\\11 = b$

Taking the Point 2

$-1 = -3(4)+b\\-1 = -12+b\\11 = b$

Now we found $b$! The equation is complete!

$y = -3x+11$