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Math  /  Algebra

QuestionFind the equation of the line that passes through the point (−8,−1)(-8, -1) and has a slope of −34-\frac{3}{4}.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through a specific point and has a specific slope. Watch out! Don't mix up the xx and yy coordinates of the point, and be careful with the negative signs!

STEP 2

1. Use the Point-Slope Form
2. Simplify to Slope-Intercept Form

STEP 3

Remember, the **point-slope form** of a linear equation is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the **slope** and (x1,y1)(x_1, y_1) is a **point** on the line.
This form is *super* useful because it lets us build a line equation directly from a point and a slope!

STEP 4

We're given the point (−8,−1)(-8, -1), so x1=−8x_1 = -8 and y1=−1y_1 = -1.
Our slope is m=−34m = -\frac{3}{4}.
Let's plug these values into the point-slope form: y−(−1)=−34(x−(−8))y - (-1) = -\frac{3}{4}(x - (-8)).

STEP 5

Now, let's simplify those double negatives: y+1=−34(x+8)y + 1 = -\frac{3}{4}(x + 8).
Much cleaner!

STEP 6

We want to get our equation into the **slope-intercept form**, which is y=mx+by = mx + b.
To do this, let's distribute the −34-\frac{3}{4} to both terms inside the parentheses: y+1=−34⋅x+(−34⋅8)y + 1 = -\frac{3}{4} \cdot x + \left( -\frac{3}{4} \cdot 8 \right).

STEP 7

Now, let's simplify that multiplication: y+1=−34x−244y + 1 = -\frac{3}{4}x - \frac{24}{4}.
Notice that 244\frac{24}{4} simplifies to **6**, so we have y+1=−34x−6y + 1 = -\frac{3}{4}x - 6.

STEP 8

To get yy by itself, we need to subtract **1** from both sides of the equation: y+1−1=−34x−6−1y + 1 - 1 = -\frac{3}{4}x - 6 - 1.

STEP 9

This simplifies to y=−34x−7y = -\frac{3}{4}x - 7.
And there we have it!
Our equation is in slope-intercept form!

STEP 10

The equation of the line that passes through the point (−8,−1)(-8, -1) and has a slope of −34-\frac{3}{4} is y=−34x−7y = -\frac{3}{4}x - 7.

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