- Calculate the slope of the line passing through \((-3, 7)\) and \((5, -5)\) using \(m = \frac{y_2 - y_1}{x_2 - x_1} = -1.5\).
- Find the y-intercept by substituting the slope and one point into \(y = mx + b\), resulting in \(b = 2.5\).
- State the final linear equation as \(y = -1.5x + 2.5\).
- Match the result with the given options to identify the correct choice as \(\boxed{A}\).
Explanation
- Analyzing the Problem and Calculating the Slope
The goal is to determine the equation of a straight line that passes through two given points, \(A(-3, 7)\) and \(B(5, -5)\). A line in the form \(y = mx + b\) is defined by its slope \(m\) and its y-intercept \(b\). To find the equation, we first calculate the slope \(m\) using the coordinates of the two points. The formula for the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the provided coordinates, we have \(m = \frac{-5 - 7}{5 - (-3)} = \frac{-12}{8} = -1.5\).
- Determining the Y-intercept and the Full Equation
Now that we have the slope \(m = -1.5\), we can substitute it into the linear equation form \(y = mx + b\) to solve for the y-intercept \(b\). We can use the coordinates of either point \(A\) or \(B\) for this step. Using point \(A(-3, 7)\):
\[7 = -1.5 \times (-3) + b\]
\[7 = 4.5 + b\]
\[b = 7 - 4.5\]
\[b = 2.5\]
Thus, the equation of the line is \(y = -1.5x + 2.5\).
- Conclusion and Verification
Comparing our derived equation \(y = -1.5x + 2.5\) with the given options:
A\(y = -1.5x + 2.5\)
B\(y = 1.5x + 2.5\)
C\(y = -1.5x + 7\)
D\(y = -0.67x + 5\)
The correct option is (A). The final answer is \(\boxed{A}\).
Examples
Understanding how to determine the equation of a line from two points is essential in fields like engineering and economics. For example, if an engineer knows the temperature at two different times and assumes a constant rate of heating, they can use this linear model to predict the temperature at any other time. Similarly, a business can track costs at two different production levels to establish a linear pricing model, allowing them to estimate future costs accurately. Mastering these algebraic relationships provides the foundation for modeling and predicting real-world trends.