A line passes through the point $(-6,3)$ and has a slope of $\frac{5}{2}$. Write an equation in slope-intercept form for this line.
The problem asks us to find the equation of a line in slope-intercept form, \(y = mx + b\), given that it passes through the point \((-6, 3)\) and has a slope of \(m = \frac{5}{2}\). We have the coordinates \(x_1 = -6\) and \(y_1 = 3\), and the slope \(m = \frac{5}{2}\). Our goal is to determine the value of the y-intercept \(b\) to complete the linear equation.
We use the point-slope form of a linear equation, which is given by \(y - y_1 = m(x - x_1)\). Substituting our known values into this formula, we get:
\[y - 3 = \frac{5}{2}(x - (-6))\]
Simplifying the expression inside the parentheses:
\[y - 3 = \frac{5}{2}(x + 6)\]
Next, we distribute the slope \(\frac{5}{2}\) to the terms inside the parentheses:
\[y - 3 = \frac{5}{2}x + \left(\frac{5}{2} \times 6\right)\]
Calculating the constant term:
\[\frac{5}{2} \times 6 = \frac{30}{2} = 15\]
So, the equation becomes:
\[y - 3 = \frac{5}{2}x + 15\]
To express the equation in slope-intercept form (\(y = mx + b\)), we need to isolate \(y\). We do this by adding \(3\) to both sides of the equation:
\[y = \frac{5}{2}x + 15 + 3\]
\[y = \frac{5}{2}x + 18\]
Thus, the equation of the line in slope-intercept form is \(\boxed{y = \frac{5}{2}x + 18}\).
Understanding how to derive the equation of a line is fundamental in various fields. For example, if a company's production cost increases at a constant rate per unit (the slope) and they have fixed monthly costs (the y-intercept), this linear equation helps model total costs. By representing these relationships as a line, businesses can predict expenses, optimize resource allocation, and plan budgets more effectively. In architecture, this is used to determine the incline of a ramp or the angle of a staircase, ensuring structural integrity and accessibility.