Write an equation of the line that passes through the given point and has the given slope.
$(1,-5)$; slope $-\frac{3}{2}$
The goal is to find the equation of a line that passes through the specific point \((1, -5)\) and has a known slope of \(m = -\frac{3}{2}\). A linear equation can be uniquely determined if we have its slope and at least one point on the line. The point-slope form \(y - y_1 = m(x - x_1)\) is a perfect tool for this, where \((x_1, y_1)\) is the given point and \(m\) is the given slope. Here, our point is \((x_1, y_1) = (1, -5)\) and the slope is \(m = -\frac{3}{2}\).
We start by substituting our values into the point-slope formula:
\[y - (-5) = -\frac{3}{2}(x - 1)\]
Simplifying the left side, we get:
\[y + 5 = -\frac{3}{2}(x - 1)\]
Now, distribute the slope \(-\frac{3}{2}\) on the right side:
\[y + 5 = -\frac{3}{2}x + \left(-\frac{3}{2}\right) \times (-1)\]
\[y + 5 = -\frac{3}{2}x + \frac{3}{2}\]
To reach the slope-intercept form (\(y = mx + b\)), subtract \(5\) from both sides:
\[y = -\frac{3}{2}x + \frac{3}{2} - 5\]
Convert \(5\) to a fraction with a denominator of 2 to combine terms:
\[5 = \frac{10}{2}\]
\[y = -\frac{3}{2}x + \left(\frac{3}{2} - \frac{10}{2}\right)\]
\[y = -\frac{3}{2}x - \frac{7}{2}\]
Thus, the equation in slope-intercept form is \(y = -\frac{3}{2}x - \frac{7}{2}\).
It is often useful to write the linear equation in the standard form \(Ax + By = C\). Starting from \(y = -\frac{3}{2}x - \frac{7}{2}\), multiply the entire equation by \(2\) to clear the fractions:
\[2y = -3x - 7\]
Add \(3x\) to both sides:
\[3x + 2y = -7\]
This is the standard form of the line passing through \((1, -5)\) with a slope of \(-\frac{3}{2}\). The final answer can be expressed in either the slope-intercept form \(y = -\frac{3}{2}x - \frac{7}{2}\) or the standard form \(3x + 2y = -7\).
In physics and engineering, identifying the linear relationship between two variables is essential. For instance, if you are modeling the velocity of an object as it decelerates over time, the slope represents the constant acceleration (or deceleration) and the point represents the velocity at a specific moment. By finding the line equation as done here, you can accurately predict the object's speed at any future point in time based on its known rate of change.