Verified Answer :

  • Identify the point \((8, -3)\) and the slope \(m = \frac{1}{4}\).
  • Use the point-slope formula \(y - y_0 = m(x - x_0)\) to substitute these values.
  • Distribute the slope and solve for \(y\) to obtain the slope-intercept form.
  • The final equation of the line is \(\boxed{y = \frac{1}{4}x - 5}\).

Explanation

  1. Analyze the given information and choose the method.

The objective is to find the equation of a line that passes through the specific point \((8, -3)\) with a given slope of \(m = \frac{1}{4}\). A linear equation in the coordinate plane defines a relationship between \(x\) and \(y\). The point-slope form, which is \(\(y - y_0 = m(x - x_0)\)\), provides an efficient way to determine the equation when a point \((x_0, y_0)\) and the slope \(m\) are known. Here, \(x_0 = 8\), \(y_0 = -3\), and \(m = \frac{1}{4}\).

  1. Apply the point-slope form.

Substitute the known values \(x_0 = 8\), \(y_0 = -3\), and \(m = \frac{1}{4}\) into the point-slope formula:
\[y - (-3) = \frac{1}{4}(x - 8)\]
Simplifying the left side, we get:
\[y + 3 = \frac{1}{4}(x - 8)\]
Now, distribute the slope \(\frac{1}{4}\) to the terms inside the parentheses:
\[y + 3 = \frac{1}{4}x - \left(\frac{1}{4} \times 8\right)\]
\[y + 3 = \frac{1}{4}x - 2\]

  1. Convert to slope-intercept form.

To express the equation in the standard slope-intercept form, \(y = mx + b\), we isolate \(y\) by subtracting 3 from both sides of the equation:
\[y = \frac{1}{4}x - 2 - 3\]
\[y = \frac{1}{4}x - 5\]
The final equation of the line is \(\boxed{y = \frac{1}{4}x - 5}\).

Examples

In urban planning, if a new road needs to be constructed passing through a specific coordinate \((8, -3)\) on a grid with a defined gradient of \(\frac{1}{4}\), engineers use the equation of a line to map its path. By calculating the linear equation, they can determine the exact elevation or position at any point along the road's stretch, ensuring precise construction and alignment within the city's infrastructure grid.

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