Problem 11 - College Algebra

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 \(y = -2x + 1\) and in the general form, the equation is \(2x + y - 1 = 0\).

Step 1: Determine Slope of Given Line

First review the equation of the given line, which is \(x - 2y - 3 = 0\). This can be rewritten in slope-intercept form \(y = mx + c\) where m is the slope and c is the y-intercept. Here the equation becomes \(y = 0.5x - 1.5\). Therefore, the slope of the given line is 0.5.

Step 2: Find the Perpendicular Slope

If two lines are perpendicular, their slopes are negative reciprocals of each other. So, find the negative reciprocal of 0.5, giving the new slope as -2.

Step 3: Write the Equation in Point-Slope Form

The point-slope form of a line equation is \(y - y1 = m(x - x1)\), where \(m\) is the slope and \((x1, y1)\) are the coordinates of the given point. Substituting \(m = -2\) and \((x1, y1) = (4, -7)\) gives the equation \(y + 7 = -2(x - 4)\).

Step 4: Simplify

Solve for \(y\) to make the equation easier to understand. Simplify the equation to: \(y = -2x + 8 - 7\). The simplified point-slope equation become: \(y = -2x + 1\).

Step 5: Write Equation in General Form

The general form of line's equation is usually written as \(Ax + By + C = 0\). Rewrite the equation in this form, we get \(2x + y - 1 = 0\).

Key Concepts

Perpendicular Lines in Algebra

Understanding how two lines can be perpendicular to each other is a fundamental concept in algebra. Perpendicular lines intersect at a right angle, or 90 degrees. The slopes of two perpendicular lines have a unique relationship: they are negative reciprocals of one another. This means, if one line has a slope \(m\), the slope of the line perpendicular to it will be \(\frac{-1}{m}\).

For instance, if we have a line with a slope of \(\frac{1}{2}\), as in the exercise provided, the perpendicular slope is the negative reciprocal, which is \(\frac{-1}{\frac{1}{2}} = -2\). When tasked with finding an equation for a line perpendicular to a given one, the first step is to determine the slope of the existing line and then flip and negate it for the desired line.

General Form Linear Equation

The general form of a linear equation provides a way to represent lines algebraically where all the terms are consolidated on one side of the equation. Typically, a general form linear equation looks like \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants. The coefficients \(A\) and \(B\) cannot both be zero since they represent the horizontal and vertical components of the line's slope, respectively.

In the solution to the problem, converting \(y = -2x + 1\) into general form requires rearranging the terms to one side: \(2x + y - 1 = 0\). It's important to note that this is not the only way to write the general form; the equation could also be multiplied by a constant to give a different look, but it would still represent the same line.

Slope-Intercept Form

The slope-intercept form is another popular way to express the equation of a line, given by \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form is especially useful because it provides immediate visual information about the line.

The slope tells us how steep the line is, and the y-intercept shows us where the line starts on the graph. In the exercise solution, after finding the perpendicular slope and using the point-slope form to plot the line through a specific point, it was simplified to the slope-intercept form of \(y = -2x + 1\), making it easier to graph and understand the relation.