Question

What is the equation of the line that has a slope of `-3/4` and passes through point (8,2) ?

Answer 1

`y = -3/4 x +8`

Explanation:

Equation of the line with slope of `-3/4` and passes through `(8,2)`.

Use the point slope formula where `m=` slope
and `(x_1, y_1)` is a point.

`y- y_1=m(x-x_1)`

`y-2= -3/4 (x-8)`

This is the equation in point slope form.

If you'd like to express the equation in slope intercept form, distribute the `-3/4` across the terms in the parentheses.

`y-2= -3/4x -3/4 * -8`

`y-2 = -3/4 x +6`

`color(white)(a)+2color(white)(aaaaaaaa)+2color(white)(aaa)`Add 2 to both sides

`y = -3/4 x +8`

Answer 2

`y = -\frac{3}{4}x + 8`

or

`3x + 4y = 32`.

Explanation:

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 + b`, where `m` is defined as the `\text{slope}` and `b` is defined as the `\text{y-intercept}`. This also happens to be the `\text{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}`. `m = -\frac{3}{4}`

`\text{2}`. Coordinates `(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 + b` as the following:

`y = -\frac{3}{4}x + b`.

Note that all I did was replace the `m` variable with the actual number. Well, we still need to find our `b` term! We can use (2) to accomplish just that. Input the coordinates `(8, 2)` into our equation. Since `x = 8` and `y = 2`, we have

`2=-\frac{3}{4} \cdot 8 + b = -\frac{24}{4} + b`

Luckily for us, 4 divides into -24 perfectly times, leaving us with -6

`2 = -6 + b`

Adding `+6` to both sides (you try this step and see what happens!), we obtain

`8 = b`.

Hence, our `\text{equation of the line}` in `\text{slope-intercept form}` is `y = -\frac{3}{4}x + 8`. We can also multiply each term by 4 to obtain

`4y = -3x + 32` which becomes `3x + 4y = 32` after we add `3x` to both sides of the equation.

`\text{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 = 8`, we should get `y = 2` since `(8, 2)` is a coordinate on the line. Well, let's plug in `x = 8` into our solution and, if we get y = 2, we definitely know we're correct.

Using the slope-intercept form and inputting `x = 8`, we obtain

`y = -\frac{3}{4} \cdot8 + 8 = -6 + 8 = 2`.

So, when `x = 8`, we get `y = 2`. This verifies that our solution is correct.

• htriber