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Math  /  Algebra

QuestionListen
Write an equation of the line that passes through (−6,0)(-6,0) and (0,−24)(0,-24).

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through two specific points. Watch out! Don't mix up the xx and yy coordinates!
Also, remember there are different ways to write the equation of a line, like slope-intercept form (y=mx+by = mx + b) and point-slope form (y−y1=m(x−x1)y - y_1 = m(x - x_1)).

STEP 2

1. Find the slope.
2. Use point-slope form.
3. Simplify to slope-intercept form.

STEP 3

Alright, let's **start** by finding the **slope**!
Remember, the slope (mm) is the change in yy divided by the change in xx.
It's like how steep a hill is!

STEP 4

We have our two points: (−6,0)(-6, 0) and (0,−24)(0, -24).
Let's label them!
Let (x1,y1)=(−6,0)(x_1, y_1) = (-6, 0) and (x2,y2)=(0,−24)(x_2, y_2) = (0, -24).
See how we're keeping things organized?

STEP 5

Now, plug those values into our **slope formula**: m=y2−y1x2−x1=−24−00−(−6) m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-24 - 0}{0 - (-6)}

STEP 6

Simplify the **numerator** and **denominator**: m=−246 m = \frac{-24}{6}

STEP 7

Divide to get our **slope**: m=−4 m = -4 So, our line goes downwards pretty steeply!

STEP 8

Now that we have our **slope**, −4-4, and we have a couple of handy points, we can use the **point-slope form** of a linear equation.
Remember, that's y−y1=m(x−x1)y - y_1 = m(x - x_1).

STEP 9

Let's use the point (−6,0)(-6, 0) as our (x1,y1)(x_1, y_1) values.
We already found that m=−4m = -4.
Let's **plug it all in**: y−0=−4(x−(−6)) y - 0 = -4(x - (-6))

STEP 10

We're almost there!
Let's simplify this to **slope-intercept form**, which is y=mx+by = mx + b.

STEP 11

Distribute the −4-4 on the right side: y=−4(x+6) y = -4(x + 6) y=−4x−24 y = -4x - 24

STEP 12

And there we have it!
Our equation in **slope-intercept form**!

STEP 13

The equation of the line is y=−4x−24y = -4x - 24.

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