What is the equation of the line that passes through the point (6, -5) and has a slope of -1/6?

Understand the Problem

The question is asking for the equation of a line that passes through a specific point, (6, -5), and has a given slope of -1/6. To solve this, we can use the point-slope form of a linear equation.

Answer

The equation of the line is $$ y = -\frac{1}{6}x - 4 $$

Steps to Solve

Identify the point and slope

The point provided is ( (6, -5) ) and the slope is ( m = -\frac{1}{6} ).

Use the point-slope form of the line

The point-slope form of the line equation is given by:

$$ y - y_1 = m(x - x_1) $$

Substituting the point ( (x_1, y_1) = (6, -5) ) and ( m = -\frac{1}{6} ):

$$ y - (-5) = -\frac{1}{6}(x - 6) $$

Simplify the equation

Rearranging the equation:

$$ y + 5 = -\frac{1}{6}(x - 6) $$

Now distribute the slope on the right side:

$$ y + 5 = -\frac{1}{6}x + 1 $$

Isolate ( y )

Subtract ( 5 ) from both sides to isolate ( y ):

$$ y = -\frac{1}{6}x + 1 - 5 $$

$$ y = -\frac{1}{6}x - 4 $$

This is the equation of the line in slope-intercept form.

The equation of the line is

$$ y = -\frac{1}{6}x - 4 $$

More Information

This equation represents a line that passes through the point ( (6, -5) ) with a slope of ( -\frac{1}{6} ). The negative slope indicates that as ( x ) increases, ( y ) decreases, typical for a downward sloping line.

Tips

Incorrectly substituting values: Ensure that the point coordinates and slope are substituted correctly into the point-slope formula.
Forgetting to distribute the slope: When moving from point-slope to slope-intercept form, do not forget to distribute the slope correctly.

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