Problem 47 - Algebra 1

Question

Write an equation of a line through \((4,5)\) that is perpendicular to \(y=\frac{1}{2} x+3\)

Step-by-Step Solution

Answer

The equation of the line passing through the point (4,5) and perpendicular to the line \(y=\frac{1}{2} x+3\) is \(y=-2x+13\).

Step 1: Calculate the Perpendicular Slope

The slope of the given line \(y=\frac{1}{2} x+3\) is 1/2. The slope of the line that is perpendicular to this line is the negative reciprocal of 1/2. This results in -2.

Step 2: Use the Point-Slope Formula

The equation of a line can be written as \(y-y1=m(x-x1)\), where m is the slope of the line and (x1, y1) is any point on the line. Here, m is -2 and (x1, y1) is (4,5). Thus, this becomes \(y-5=-2(x-4)\).

Step 3: Simplify the Equation

Distribute the -2: \(y-5=-2x+8\).

Step 4: Write the Line in Slope-Intercept Form

Rearrange the equation to be in the format y=mx+b. This results in \(y=-2x+13\).

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. It is calculated as the "rise over run." This means you divide the change in the y-values by the change in the x-values between two points on the line. The formula for the slope, often denoted as \(m\), can be given as:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In the context of perpendicular lines, if you know the slope of one line, the slope of a line perpendicular to it will be its negative reciprocal. For example, if one line has a slope of \(1/2\), the perpendicular line will have a slope of \(-2\). This rule helps us find equations of lines that are perpendicular to each other.

Point-Slope Formula

The point-slope formula is a useful way to write the equation of a line when you know a point on the line and its slope. The point-slope equation is written as:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) represents the slope, and \((x_1, y_1)\) represents a point on the line. This formula is particularly handy for quickly forming an equation when the slope and a specific point are given. In our exercise, we applied the point-slope formula with the slope \(-2\) and the point \((4,5)\) to build the line equation:

• \(y - 5 = -2(x - 4)\)

This formula helps simplify the process of writing linear equations.

Slope-Intercept Form

The slope-intercept form is one of the most familiar ways to express the equation of a line. It is given by:

• \(y = mx + b\)

In this form, \(m\) is the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. Converting an equation to slope-intercept form makes it easy to graph and understand the line's properties at a glance.
The exercise involved transforming the equation from point-slope form to slope-intercept form:

• Start with \(y - 5 = -2(x - 4)\)
• Distribute and simplify to get \(y = -2x + 13\)

This shows a neat slope of \(-2\) and a y-intercept of \(13\), allowing you to quickly graph and analyze the line.