Complete the equation of the line through $(-9,7)$ and $(-6,-3)$. Use exact numbers.
$y=$
We are given two points on a line: \(P_1 = (-9, 7)\) and \(P_2 = (-6, -3)\). Our objective is to determine the equation of the line passing through these points in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To find this equation, we first calculate the slope \(m\) and then use one of the given points to solve for the y-intercept \(b\).
The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the coordinates \(P_1(-9, 7)\) and \(P_2(-6, -3)\) into the formula, we get:
\[m = \frac{-3 - 7}{-6 - (-9)}\]
\[m = \frac{-10}{-6 + 9}\]
\[m = \frac{-10}{3}\]
Thus, the slope of the line is \(m = -\frac{10}{3}\).
Now that we have the slope, we can use the point-slope form \(y - y_1 = m(x - x_1)\) to find the equation. Let us use point \(P_1(-9, 7)\):
\[y - 7 = -\frac{10}{3}(x - (-9))\]
\[y - 7 = -\frac{10}{3}(x + 9)\]
Next, we expand the right side:
\[y - 7 = -\frac{10}{3}x - \left(\frac{10}{3} \times 9\right)\]
\[y - 7 = -\frac{10}{3}x - 30\]
Finally, we add 7 to both sides to solve for \(y\):
\[y = -\frac{10}{3}x - 30 + 7\]
\[y = -\frac{10}{3}x - 23\]
The y-intercept \(b\) is \(-23\).
We have determined the slope \(m = -\frac{10}{3}\) and the y-intercept \(b = -23\). The equation of the line is:
\[y = -\frac{10}{3}x - 23\]
This equation satisfies both original points. For \(x = -9\), \(y = -\frac{10}{3}(-9) - 23 = 30 - 23 = 7\). For \(x = -6\), \(y = -\frac{10}{3}(-6) - 23 = 20 - 23 = -3\). The result is correct.
\[y = \boxed{-\frac{10}{3}x - 23}\]
In geography or logistics, finding a linear equation between two points is essential for calculating travel paths or mapping routes. For example, if you know the elevation of two different locations along a mountain trail, you can find the linear slope to predict the steepness at any other point on that segment. Understanding how to derive this equation helps in planning infrastructure, estimating fuel consumption, or determining the shortest path between two points in a coordinate system.