- Calculate the slope of the line using the points \((-8, 11)\) and \((4, \frac{7}{2})\), which results in \(m = -\frac{5}{8}\).
- Apply the point-slope form \(y - y_1 = m(x - x_1)\) to derive the equation in slope-intercept form.
- Simplify the expression to reach the final equation \(y = -\frac{5}{8}x + 6\).
- Conclude that the correct option is \(\boxed{y = -\frac{5}{8}x + 6}\).
Explanation
- Calculating the Slope of the Line
To find the equation of the line passing through the points \(P_1 = (-8, 11)\) and \(P_2 = (4, \frac{7}{2})\), we first determine the slope \(m\) of the line. The slope formula is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given coordinates \(x_1 = -8, y_1 = 11\) and \(x_2 = 4, y_2 = \frac{7}{2}\) (which is \(3.5\)):
\[m = \frac{3.5 - 11}{4 - (-8)} = \frac{-7.5}{12}\]
We can simplify this fraction:
\[m = \frac{-7.5}{12} = \frac{-75}{120} = -\frac{5}{8}\]
So, the slope of the line is \(-\frac{5}{8}\).
- Determining the Equation of the Line
Now that we have the slope \(m = -\frac{5}{8}\), we use the point-slope form of the linear equation, which is \(y - y_1 = m(x - x_1)\), to find the equation of the line. We can use the point \((-8, 11)\) as \((x_1, y_1)\):
\[y - 11 = -\frac{5}{8}(x - (-8))\]
\[y - 11 = -\frac{5}{8}(x + 8)\]
Distribute the slope on the right side:
\[y - 11 = -\frac{5}{8}x - \left(\frac{5}{8} \times 8\right)\]
\[y - 11 = -\frac{5}{8}x - 5\]
Now, solve for \(y\) by adding 11 to both sides:
\[y = -\frac{5}{8}x - 5 + 11\]
\[y = -\frac{5}{8}x + 6\]
The equation of the line is \(y = -\frac{5}{8}x + 6\). Comparing this to the given options, it corresponds to the first option.
Examples
Linear equations are fundamental for modeling real-world relationships where one variable changes proportionally to another. For instance, if a delivery company charges a fixed base fee of \(\$6\) and an additional cost of \(\$0.625\) (which is \(-\frac{5}{8}\) if we think in terms of reduction of some resource) per mile for a specific service, this equation represents the total cost or resource usage over time or distance. Mastering this helps in planning budgets, understanding physical motion, or analyzing data trends where two variables have a linear dependency.