Problem 11 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through \((4,-7)\) and perpendicular to the line whose equation is \(x-2 y-3=0\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 7 = -2x + 8\). In general form, the equation of the line is \(2x + y - 15 = 0.\)

Step 1: Find the Slope of the Given Line

From the equation \(x-2y-3 = 0\), we can rewrite it in the form \(y = mx + c\) to find the slope. Therefore, the rearranged equation is \(2y = x - 3\) or \(y=(x/2) - 3/2\). The slope (m) of this line is 1/2.

Step 2: Find the Slope of the Perpendicular Line

The slope (m') of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of the required line will be \(m' = -1/m = -1/(1/2) = -2.\)

Step 3: Write the Equation in Point-Slope Form

The point-slope form of a line is \(y - y_1 = m'(x - x_1)\). We know that the line passes through the point (4,-7). Substituting these values along with the slope m' = -2 into the formula, the equation becomes \(y - (-7) = -2(x - 4)\), Simplifying it, we get \(y + 7 = -2x + 8\)

Step 4: Convert to General Form

The general form of a line is \(Ax + By = C\). By arranging the equation obtained from the point-slope form, we get \(2x + y - 15 = 0\)