- Identify the given point \((x_1, y_1) = (-9, 3)\) and slope \(m = 1\).
- Apply the point-slope form equation: \(y - y_1 = m(x - x_1)\).
- Substitute the values to obtain: \(y - 3 = 1(x - (-9))\).
- Simplify the expression to arrive at the final result: \(\boxed{y - 3 = 1(x + 9)}\).
Explanation
- Analyzing the Problem Data
To find the equation of a line in point-slope form, we start with the given data. We are provided with a specific point that the line passes through, \((x_1, y_1) = (-9, 3)\), and the slope of the line, \(m = 1\). The point-slope form of a linear equation is a fundamental tool for representing lines on a coordinate plane.
- Recalling the Point-Slope Formula
The point-slope form of a linear equation is defined as:
\[y - y_1 = m(x - x_1)\]
This formula allows us to write the equation of a line once we know any point \((x_1, y_1)\) on the line and the slope \(m\). We have all the necessary components for this equation already.
- Substituting and Simplifying the Equation
Now, we substitute the known values \(x_1 = -9\), \(y_1 = 3\), and \(m = 1\) into the formula:
\[y - 3 = 1(x - (-9))\]
Next, we simplify the term inside the parentheses. Subtracting a negative is equivalent to adding a positive, so \(x - (-9)\) becomes \(x + 9\). Thus, the equation becomes:
\[y - 3 = 1(x + 9)\]
This is the equation of the line in point-slope form, using the point \((-9, 3)\) as required.
Examples
The point-slope form is essential in fields like physics and engineering, where you often need to model relationships between variables based on a known data point and a constant rate of change. For example, if you are tracking the speed of a vehicle that passes a specific point at a certain time and you know it is accelerating at a constant rate, you can use the point-slope form to predict its position at any other time. This simple algebraic representation acts as a foundation for understanding how linear systems behave in the real world.