Math / Geometry

QuestionWhat is the slope of the line that passes through th points $(3,-3)$ and $(9,-6)$ ?

Studdy Solution

STEP 1

What is this asking? We need to find how steep a line is that goes through two specific points. Watch out! Don't mix up the $x$ and $y$ coordinates!
Also, remember that a negative slope means the line goes downhill from left to right.

STEP 2

1. Find the change in $y$ .
2. Find the change in $x$ .
3. Calculate the slope.

STEP 3

Let's call our points Point 1 $(x_1, y_1)$ and Point 2 $(x_2, y_2)$ .
So, for Point 1, we have $x_1 = 3$ and $y_1 = -3$ .
For Point 2, we have $x_2 = 9$ and $y_2 = -6$ .

STEP 4

The change in $y$ is the difference between the $y$ -coordinates, which we calculate as $y_2 - y_1$ .
Plugging in our values, we get $(-6) - (-3)$ .

STEP 5

Remember, subtracting a negative is the same as adding a positive!
So, $(-6) - (-3)$ becomes $-6 + 3 = -3$ .
This means our change in $y$ is -3.

STEP 6

Similar to what we did with the $y$ -coordinates, the change in $x$ is $x_2 - x_1$ .
Using our values, this is $9 - 3 = 6$ .
So, our change in $x$ is 6.

STEP 7

The slope of a line, often denoted by $m$ , is defined as the change in $y$ divided by the change in $x$ .
We can write this as a formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 8

We already found that the change in $y$ is $-3$ and the change in $x$ is $6$ .
Let's plug these values into our slope formula: $m = \frac{-3}{6}$

STEP 9

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is $3$ .
This gives us: $m = \frac{-3 \div 3}{6 \div 3} = \frac{-1}{2}$

STEP 10

The slope of the line passing through the points $(3, -3)$ and $(9, -6)$ is $-\frac{1}{2}$ .

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