Write the equation of the line that passes through the points (−9,−9) and (−8,1). 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 the points (-9,-9) and (-8,1), we can use the point-slope form of a linear equation:
\(\displaystyle{ y−y_{1}=m(x−x_{1})}\)
Where \(m\) is the slope of the line, and \((x_{1},y_{1})\) are the coordinates of one of the given points.
First, let's calculate the slope of the line using the formula:
\(\displaystyle{ m=\frac{{y_{2}−y_{1}}}{{x_{2}−x_{1}}}}\)
By substituting the values \(\displaystyle{ x_{1}=-9,\ y_{1}=-9,\ x_{2}=-8,\ y_{2}=1}\) into the slope formula, we get:
\(\displaystyle{ m=\frac{{1−(−9)}}{{(−8)−(−9)}}}\)
\(\displaystyle{ m=\frac{{1+9}}{{−8+9}}}\)
\(\displaystyle{ m=\frac{{10}}{{1}}}\)
\(\displaystyle{ m=10}\)
Therefore, the slope of the line passing through the given points is 10.
Now that we have the slope, we can use one of the given points, \((-9,-9)\), and the slope in the point-slope form:
\(\displaystyle{ y−y_{1}=m(x−x_{1})}\)
Substituting \(\displaystyle{ x_{1}=-9,\ y_{1}=-9,\ m=10}\), we have:
\(\displaystyle{ y−(-9)=10(x−(-9))}\)
\(\displaystyle{ y+9=10(x+9)}\)
Expanding the right side, we obtain:
\(\displaystyle{ y+9=10x+90}\)
Finally, let's fully simplify the equation:
\(\displaystyle{ y=10x+81}\)
Thus, the equation of the line passing through the points \((-9,-9)\) and \((-8,1)\) in fully simplified point-slope form is \(y=10x+81\).