Math  /  Algebra

QuestionWhat is the equation of the line that passes through the point (6,8)(-6,-8) and has a slope of 12\frac{1}{2} ?
Answer Attempt 1 out of 2

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a specific line that goes through a given point and has a specific slope. Watch out! Don't mix up the xx and yy coordinates of the point!
Also, remember that the slope-intercept form is y=mx+by = mx + b, not y=bx+my = bx + m.

STEP 2

1. Use the point-slope form.
2. Convert to slope-intercept form.

STEP 3

The **point-slope form** of a linear equation is super useful when we know a point (x1,y1)(x_1, y_1) on the line and the slope mm of the line.
It's given by yy1=m(xx1)y - y_1 = m(x - x_1).
It's like a shortcut to the line's equation!

STEP 4

Our point is (6,8)(-6, -8), so x1=6x_1 = -6 and y1=8y_1 = -8.
Our slope is m=12m = \frac{1}{2}.
Let's plug these values into the point-slope form: y(8)=12(x(6))y - (-8) = \frac{1}{2}(x - (-6)).

STEP 5

Subtracting a negative is the same as adding, so we have y+8=12(x+6)y + 8 = \frac{1}{2}(x + 6).

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 12\frac{1}{2} to both terms inside the parentheses: y+8=12x+126y + 8 = \frac{1}{2} \cdot x + \frac{1}{2} \cdot 6, which simplifies to y+8=12x+3y + 8 = \frac{1}{2}x + 3.

STEP 7

To get yy by itself, we need to subtract 88 from both sides of the equation: y+88=12x+38y + 8 - 8 = \frac{1}{2}x + 3 - 8.

STEP 8

This gives us our final equation in slope-intercept form: y=12x5y = \frac{1}{2}x - 5.

STEP 9

The equation of the line that passes through the point (6,8)(-6, -8) and has a slope of 12\frac{1}{2} is y=12x5y = \frac{1}{2}x - 5.

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