Write the equation of the line that passes through the points left bracket, minus, 1, comma, 3, right bracket(−1,3) and left bracket, minus, 9, comma, 5, right bracket(−9,5). Put your answer in fully simplified point-slope form, unless it is a vertical or horizontal line.
To find the equation of a line in point-slope form, we need the slope of the line and one point on the line. We can find the slope using the formula:
\[m = \dfrac{y_2 - y_1}{x_2 - x_1}\]
Using the points \((-1,3)\) and \((-9,5)\), we can substitute the values into the formula:
\[m = \dfrac{5 - 3}{-9 - (-1)}\]
\[m = \dfrac{2}{-8} = -\dfrac{1}{4}\]
So the slope of the line is \(-\dfrac{1}{4}\).
Now we can use the point-slope form to write the equation of the line. The point-slope form is
\[y - y_1 = m(x - x_1)\]
Using the point \((-1,3)\) and the slope \(-\dfrac{1}{4}\), we substitute the values into the equation:
\[y - 3 = -\dfrac{1}{4}(x - (-1))\]
\[y - 3 = -\dfrac{1}{4}(x + 1)\]
To simplify the equation, we distribute the \(-\dfrac{1}{4}\) to the terms inside the parentheses:
\[y - 3 = -\dfrac{1}{4}x - \dfrac{1}{4}\]
Finally, we can write the equation of the line in fully simplified point-slope form as:
\[y = -\dfrac{1}{4}x - \dfrac{1}{4} + 3\]
\[y = -\dfrac{1}{4}x + \dfrac{11}{4}\]