A line passes through $(1,1)$ and $(-3,5)$. What is the equation of the line in slope-intercept form?
To find the equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope-intercept form equation \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Given the points \(P_1 = (1, 1)\) and \(P_2 = (-3, 5)\), our task is to calculate the values of \(m\) and \(b\).
The slope \(m\) is calculated using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}.\] Substituting the coordinates of our points, \(P_1 = (1, 1)\) and \(P_2 = (-3, 5)\), we get: \[m = \frac{5 - 1}{-3 - 1} = \frac{4}{-4} = -1.\] Thus, the slope of the line is \(-1\).
Now that we have the slope \(m = -1\), we can substitute it into the slope-intercept form \(y = mx + b\) to solve for the y-intercept \(b\). Using point \(P_1 = (1, 1)\), we have: \[1 = (-1)(1) + b\] \[1 = -1 + b\] Adding 1 to both sides gives: \[b = 2\] Therefore, the y-intercept is 2.
With the slope \(m = -1\) and the y-intercept \(b = 2\), the equation of the line is \[y = -1x + 2, \text{ or simply } y = -x + 2.\] Comparing this to the given options, we find it corresponds to the first option: \(y = -x + 2\). The final answer is \(\boxed{y = -x + 2}\).
Finding the equation of a line is fundamental in various real-life applications. For instance, if you are tracking the fuel consumption of a car over time, you might plot the distance traveled versus the remaining fuel. By identifying the line that passes through two specific data points (e.g., at 100 miles you have 10 gallons left, and at 200 miles you have 5 gallons left), you can derive a linear equation \(y = mx + b\) to predict how much fuel you will have at any given distance, ensuring you don't run out of gas. This skill is essential for planning, budgeting, and forecasting trends.