Problem 83 - College Algebra

Question

If one point on a line is \((3,-1)\) and the line's slope is \(-2,\) find the \(y\) -intercept.

Step-by-Step Solution

Answer

The y-intercept of the line is 5. Therefore, when the line is drawn on a graph, it will cross the y-axis at the point (0,5).

Step 1: Identify given values

We know that the slope \(m = -2\) and a point \((x_1,y_1) = (3,-1)\) on the line.

Step 2: Substitute into point-slope formula

Substitute \(m, x_1,\) and \(y_1\) into the point-slope form \(y - y_1 = m(x - x_1)\). This gives us \(y - (-1) = -2(x - 3)\). After simplifying, we find \(y + 1 = -2x + 6\).

Step 3: Solve for y-intercept

To isolate \(y\), we subtract 1 from both sides of the equation, resulting in \(y = -2x + 5\). In a linear function of the form \(y = mx + b,\) the y-intercept is the constant \(b.\) Therefore, the y-intercept of this function is \(5.\)

Step 4: Final check

The equation of the line, with the found \(y\)-intercept, is \(y = -2x + 5\). The \(y\)-intercept is the point where the line crosses the \(y\)-axis. This happens when \(x=0\). If we substitute \(x=0\) in the equation, we will find \(y=5\). Therefore, the solution is correct.