Problem 84 - College Algebra

Question

If one point on a line is \((2,-6)\) and the line's slope is \(-\frac{3}{2},\) find the \(y\) -intercept.

Step-by-Step Solution

Answer

The y-intercept of the line is -3.

Step 1: Substitute the given point and slope into the point-slope form of the equation

Plug the given values into \(y - y_1 = m(x - x_1)\). In this case, \(x_1 = 2\), \(y_1 = -6\), and \(m = -3/2\). The equation then changes to \(y - (-6) = -3/2 (x-2)\). Which simplifies to \(y + 6 = -3/2x + 3\).

Step 2: Convert the equation to slope-intercept form

Rearrange the equation from step 1 into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Subtract 6 from both sides to solve for \(y\). The equation then becomes \(y = -3/2x + 3 - 6\). This simplifies to \(y = -3/2x -3\).

Step 3: Identify the y-intercept

In the equation \(y = -3/2x -3\), the coefficient of \(x\) is the slope, and the constant term is the y-intercept. So, it can be concluded that the y-intercept is -3.