Problem 85 - Algebra 2

Question

Which equation represents a line through \((3,5)\) that is perpendicular to \(y=2 x-5 ?\) $$ \begin{array}{ll}{\text { A. } 2 y=-x+13} & {\text { B. } 2 y=x+13} \\ {\text { C. } 2 y-x=13} & {\text { D. } 2 y+x=-13}\end{array} $$

Step-by-Step Solution

Answer

The equation representing a line through (3,5) perpendicular to \(y=2x-5\) is option C, \(2y-x=13\).

Step 1: Identify the Slope of the Given Line

The given equation of the line is in the slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope. For the equation \(y = 2x - 5\), the slope \(m\) is 2.

Step 2: Calculate the Perpendicular Slope

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. As our original slope is 2, the perpendicular slope will be \(-1/2\).

Step 3: Create the Equation of the Perpendicular Line

The equation of the line perpendicular to the given line with slope \(-1/2\) that passes through point \((3,5)\) is found using the point-slope form, which is \(y - y_1 = m(x - x_1)\). Plugging the values in, we get \(y - 5 = -\frac{1}{2}(x - 3)\).

Step 4: Simplify and Rewrite in Standard Form

Rewrite the equation to match the options: multiply all terms by 2 to eliminate fractions and bring the terms to one side to arrive at \(2y = -x + 13\).

Step 5: Match the Equation to the Provided Options

The equation we found, \(2y = -x + 13\), needs to be correctly ordered to match one of the multiple-choice options. Rearrange to get the option that matches: \(2y + x = 13\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is key when dealing with linear equations. It's written as (y = mx + b), where m represents the slope of the line and b indicates the y-intercept – the point where the line crosses the y-axis.

In our exercise, the original equation (y = 2x - 5) is already in slope-intercept form. The slope here is 2, and the y-intercept is -5. One crucial thing to remember is that when two lines are perpendicular, their slopes are negative reciprocals of each other. This means you can quickly identify the slope of the perpendicular line once you know the slope of the initial line.

Point-Slope Form

When you have a point through which a line passes and the slope of that line, the point-slope form comes in handy. The formula is written as (y - y_1 = m(x - x_1)), where (x_1, y_1) is the point on the line and m is the slope. This can be a more direct method for finding the equation of a line than slope-intercept form, especially when finding lines parallel or perpendicular to a given line.

For the exercise, we used the point-slope form to create an equation for a line perpendicular to the original through point (3,5). After plugging the perpendicular slope and the point into the formula, we derived the perpendicular line's equation, illustrating the utility of point-slope form in such situations.

Negative Reciprocal Slope

The concept of negative reciprocal slope is vital in understanding the relationship between perpendicular lines. If one line's slope is (m), the slope of a line perpendicular to it will be (-1/m). It's the product of the slopes that equals -1 which is the defining characteristic of perpendicular lines in the Cartesian plane.

In the original equation, the slope of the line given was 2. To find the slope of the perpendicular line, take the negative reciprocal, resulting in -1/2. This new slope was used to determine the correct equation of the line that is perpendicular to the original, through the point (3,5), illustrating how the reciprocal nature of the slope dictates the orientation of perpendicular lines.