QuestionUse the given conditions to write an equation for the line in point-slope form. 12) Passing through $(8, 8)$ and $(6, 5)$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two specific points, and we want that equation in point-slope form. Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember point-slope form is $y - y_1 = m(x - x_1)$ , not $y = mx + b$ .

STEP 2

1. Find the slope.
2. Write the equation.

STEP 3

Alright, let's start by finding the slope!
The slope, which we usually call $m$ , tells us how steep our line is.
The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 4

Our two points are $(8, 8)$ and $(6, 5)$ .
Let's label these points to avoid confusion!
We'll call $(8, 8)$ our $(x_1, y_1)$ and $(6, 5)$ our $(x_2, y_2)$ .
So, $x_1 = 8$ , $y_1 = 8$ , $x_2 = 6$ , and $y_2 = 5$ .

STEP 5

Now, let's plug these values into our slope formula: $m = \frac{5 - 8}{6 - 8}$

STEP 6

Calculate the differences: $m = \frac{-3}{-2}$

STEP 7

Simplify the fraction by dividing both numerator and denominator by $-1$ : $m = \frac{3}{2}$ So, our slope is $\frac{3}{2}$ !

STEP 8

Now that we have our slope, we can use the point-slope form of a linear equation, which is: $y - y_1 = m(x - x_1)$

STEP 9

We already have our slope $m = \frac{3}{2}$ and we can use either of our points for $(x_1, y_1)$ .
Let's use $(8, 8)$ , so $x_1 = 8$ and $y_1 = 8$ .

STEP 10

Substitute these values into the point-slope form: $y - 8 = \frac{3}{2}(x - 8)$ And there we have it!

STEP 11

The equation of the line in point-slope form passing through $(8, 8)$ and $(6, 5)$ is $y - 8 = \frac{3}{2}(x - 8)$ .

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