- Identify the two points \((-12, 14)\) and \((6, -1)\) to calculate the slope using \(m = \frac{y_2 - y_1}{x_2 - x_1} = -\frac{5}{6}\).
- Utilize the point-slope form \(y - y_1 = m(x - x_1)\) to set up the linear equation.
- Simplify the expression to reach the slope-intercept form \(y = mx + b\).
- Verify the result using the second point to obtain the final equation \(\boxed{y = -\frac{5}{6}x + 4}\).
Explanation
- Analysis and Problem Setup
The goal is to find the equation of a line passing through the two points \(P_1 = (-12, 14)\) and \(P_2 = (6, -1)\). A line is uniquely determined by these two points, and its equation can be represented in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We will first calculate the slope of the line, then use the point-slope form to derive the final equation.
- Calculating the Slope
The slope \(m\) of a line passing through \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given points \(P_1 = (-12, 14)\) and \(P_2 = (6, -1)\):
\[m = \frac{-1 - 14}{6 - (-12)} = \frac{-15}{6 + 12} = \frac{-15}{18}\]
By simplifying the fraction by dividing the numerator and denominator by 3, we get:
\[m = -\frac{5}{6}\]
- Finding the Slope-Intercept Form
Now that we have the slope \(m = -\frac{5}{6}\), we can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\). Using point \(P_1 = (-12, 14)\):
\[y - 14 = -\frac{5}{6}(x - (-12))\]
\[y - 14 = -\frac{5}{6}(x + 12)\]
Next, distribute the slope on the right side:
\[y - 14 = -\frac{5}{6}x - \frac{5}{6} \times 12\]
\[y - 14 = -\frac{5}{6}x - 10\]
Add 14 to both sides to solve for \(y\):
\[y = -\frac{5}{6}x - 10 + 14\]
\[y = -\frac{5}{6}x + 4\]
- Verification and Conclusion
To ensure the accuracy of the result, we can check the equation \(y = -\frac{5}{6}x + 4\) by substituting the coordinates of the second point \(P_2 = (6, -1)\):
\[y = -\frac{5}{6}(6) + 4 = -5 + 4 = -1\]
Since the coordinates \((6, -1)\) satisfy the equation, the derived linear equation is correct. The final equation of the line is \(\boxed{y = -\frac{5}{6}x + 4}\).
Examples
Finding the equation of a line is fundamental in various real-life scenarios, such as predicting trends. For example, if you are tracking the distance of a train from a station over time, the points could represent its position at different times. By establishing the linear equation \(y = mx + b\), you can predict exactly when the train will reach a certain destination or determine how fast it was moving (the slope). Understanding these relationships allows you to model simple, constant-speed motions accurately in everyday logistics and travel planning.