- Identify the slope \(m = 5\) from the equation \(y = 5x - 3\).
- Use the point \((-2, -13)\) as \((x_1, y_1)\).
- Substitute these into the point-slope formula \(y - y_1 = m(x - x_1)\).
- Simplify to get the final form \(\boxed{y + 13 = 5(x + 2)}\).
Explanation
- Analyzing the Problem
We are given that a line passes through the point \((-2, -13)\) and has a slope-intercept form equation of \(y = 5x - 3\). Our objective is to find the point-slope form of this line equation, which is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a specific point on the line.
- Identifying Parameters
From the provided slope-intercept equation \(y = 5x - 3\), we can identify the slope \(m\) as the coefficient of \(x\), which is \(5\). The point \((x_1, y_1)\) through which the line passes is given as \((-2, -13)\). So, we have \(m = 5\), \(x_1 = -2\), and \(y_1 = -13\).
- Deriving the Point-Slope Form
Substituting the values \(m = 5\), \(x_1 = -2\), and \(y_1 = -13\) into the point-slope formula \(y - y_1 = m(x - x_1)\), we get:
\[y - (-13) = 5(x - (-2))\]
Simplifying the signs within the parentheses, we get:
\[y + 13 = 5(x + 2)\]
This expression matches the second option provided.
- Conclusion
The point-slope form of the line passing through \((-2, -13)\) with a slope of \(5\) is \(y + 13 = 5(x + 2)\). This is the second option listed in the problem. \(\boxed{y + 13 = 5(x + 2)}\).
Examples
Point-slope form is highly practical in engineering and architecture when designing sloped surfaces like roofs or ramps. If you know the exact coordinates of one point on a roof edge and the steepness (slope) required for drainage, you can easily derive the equation of that line to model the entire slope. This mathematical representation allows for precise construction measurements and material estimates.