Line $C D$ passes through points $C(3,-5)$ and $D(6,0)$. What is the equation of line $C D$ in standard form?
We are given two points \(C(3, -5)\) and \(D(6, 0)\) that lie on a line. Our goal is to find the equation of the line passing through these points in standard form, \(Ax + By = C\), where \(A, B, C\) are constants. The coordinates of the points are \((x_1, y_1) = (3, -5)\) and \((x_2, y_2) = (6, 0)\).
The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the given coordinates:
\[m = \frac{0 - (-5)}{6 - 3} = \frac{5}{3}\]
So, the slope of the line is \(\frac{5}{3}\).
We can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\). Using point \(D(6, 0)\) and the slope \(m = \frac{5}{3}\):
\[y - 0 = \frac{5}{3}(x - 6)\]
\[y = \frac{5}{3}x - \frac{5}{3} \times 6\]
\[y = \frac{5}{3}x - 10\]
To convert the equation into the standard form \(Ax + By = C\), we can multiply the entire equation by 3 to eliminate the fraction:
\[3y = 5x - 30\]
Next, we rearrange the terms to have \(x\) and \(y\) on the left side:
\[-5x + 3y = -30\]
Multiplying by -1 to keep the leading coefficient positive, we obtain:
\[5x - 3y = 30\]
This matches the second option provided.
In urban planning, identifying the path of a road or a utility line that must pass through two specific locations is a fundamental task. By calculating the linear equation of the path, engineers can determine the exact coordinates for the infrastructure and ensure it correctly connects the desired points. This mathematical modeling allows for precise design and efficient resource distribution in city development projects.