Problem 45 - Calculus with Concepts in Calculus

Question

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: 2 x+3 y-6=0 ; P=(2,3)\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{3}{2}x\).

Step 1: Understand the given line

The equation provided is that of a line: \(2x + 3y - 6 = 0\). We can re-arrange it in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Step 2: Convert to slope-intercept form

Rearrange the equation: \(3y = -2x + 6\) and then divide each term by 3 to solve for \(y\): \(y = -\frac{2}{3}x + 2\). So, the slope \(m\) of line \(l\) is \(-\frac{2}{3}\).

Step 3: Determine the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Given the slope of line \(l\) is \(-\frac{2}{3}\), the slope of the perpendicular line will be \(\frac{3}{2}\).

Step 4: Use point-slope form to find the equation

We use the point-slope formula: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through. With point \(P = (2,3)\) and slope \(m = \frac{3}{2}\), the equation is \(y - 3 = \frac{3}{2}(x - 2)\).

Step 5: Simplify to slope-intercept form

Distribute \(\frac{3}{2}\): \(y - 3 = \frac{3}{2}x - 3\), then add 3 to both sides to solve for \(y\): \(y = \frac{3}{2}x\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation of a line. In this form, the equation looks like: \[ y = mx + b \] In this equation, \( m \) represents the slope of the line. The slope indicates how steep the line is. The \( b \) represents the y-intercept, or where the line crosses the y-axis.

• slope \( m \): tells us how much \( y \) will change if \( x \) increases by 1.
• y-intercept \( b \): the y-coordinate when \( x = 0 \).

This form is very useful for quickly understanding and graphing a line. You can immediately see the direction and angle of the line just from these two parameters: \( m \) and \( b \). In our example, the line \( l: 2x + 3y - 6 = 0 \) was converted to slope-intercept form \[ y = -\frac{2}{3}x + 2 \], showing that the slope \( m \) is \(-\frac{2}{3}\), and the y-intercept \( b \) is 2.

Slope Calculation

Calculating the slope of a line is fundamental to understanding linear equations. The slope is a measure of the steepness and direction of a line. It is often denoted by \( m \), and can be calculated when you have two points on the line, as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
In the case of line \( l \), we didn't need to calculate the slope from points, as it could be derived directly from its equation in slope-intercept form. Once we have the slope of a given line, finding the slope of a line that is perpendicular to it is simple, by taking the negative reciprocal. This means if the slope of line \( l \) is \(-\frac{2}{3}\), the slope of a line perpendicular to it will be \(\frac{3}{2}\). Calculating slopes this way is essential for solving many geometric problems about lines.

Point-Slope Form

The point-slope form is another way to write the equation of a line. It is especially useful when you know a point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] Where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. This form makes it easy to build the equation of a line when these conditions are met. For our task of finding the equation of a line perpendicular to \( l \) and passing through point \( P(2, 3) \), we used the point-slope form.
Given that the slope \( m = \frac{3}{2} \) and \( (x_1, y_1) = (2, 3) \), the equation becomes:\[ y - 3 = \frac{3}{2}(x - 2) \]This mathematical format gives a straightforward way to derive the full equation of a line, especially useful when point and slope information is readily available as demonstrated in finding equations of perpendicular lines.