Write the equation of the line that passes through the points (0, 8) and (6, -2). Put your answer in fully simplified point-slope form, unless it is a vertical or horizontal line.
To find the equation of the line passing through two points, we first need to find the slope of the line.
Let's use the formula for slope:
\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
Using the points (0, 8) and (6, -2), we have:
\[m = \frac{{-2 - 8}}{{6 - 0}} = \frac{{-10}}{{6}} = -\frac{5}{3}\]
Now, we can use the point-slope form of a linear equation given a slope \(m\) and a point \((x_1, y_1)\):
\[y - y_1 = m(x - x_1)\]
Using the point (0, 8):
\[y - 8 = -\frac{5}{3}(x - 0)\]
Simplifying, we get the equation in point-slope form as:
\[y - 8 = -\frac{5}{3}x\]
However, the question asks for the equation to be put in fully simplified point-slope form, unless it is a vertical or horizontal line. Since the given line is neither vertical nor horizontal, we can further simplify the equation.
Multiplying through by \(3\) to get rid of the fraction:
\[3(y - 8) = -5x\]
Distributing on the left side:
\[3y - 24 = -5x\]
Finally, rearranging the equation to be in slope-intercept form ( \(y = mx + b\) ):
\[3y = -5x + 24\]
Dividing through by \(3\):
\[y = -\frac{5}{3}x + \frac{24}{3} = -\frac{5}{3}x + 8\]
Therefore, the equation of the line passing through the points (0, 8) and (6, -2) is \(y = -\frac{5}{3}x + 8\).