Problem 46 - 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 \(P\). \(l: 3 x-y=0 ; P=(1,3)\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{1}{3}x + \frac{10}{3}\).

Step 1: Find the Slope of the Given Line

The equation of the line is given in standard form: \(3x - y = 0\). We need to express it in the slope-intercept form \(y = mx + b\) to find the slope. Solving for \(y\), we get: \(y = 3x\). Thus, the slope \(m\) of line \(l\) is \(3\).

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope \(m'\) of the line perpendicular to line \(l\) is \(-\frac{1}{3}\).

Step 3: Use the Point-Slope Form to Find the Equation

Utilize the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. For this problem, use \((x_1, y_1) = (1, 3)\) and slope \(-\frac{1}{3}\). The equation becomes: \(y - 3 = -\frac{1}{3}(x - 1)\).

Step 4: Simplify the Equation

Simplify the point-slope equation to the slope-intercept form. Multiply everything out: \(y - 3 = -\frac{1}{3}x + \frac{1}{3}\). Add \(3\) to both sides to obtain \(y = -\frac{1}{3}x + \frac{1}{3} + 3\). Simplifying further gives us: \(y = -\frac{1}{3}x + \frac{10}{3}\).

Key Concepts

Slope

The slope is a key concept in understanding how steep a line is on a graph. Think of it as a measure of how much the line rises or falls as you move from left to right along the x-axis. In mathematical terms, the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It is often represented by the letter "m." To find the slope from an equation, we convert the equation into the slope-intercept form, which looks like this: \( y = mx + b \). Here, \( m \) is the slope. In the given problem, the line \( 3x - y = 0 \) was converted to \( y = 3x \), revealing a slope \( m = 3 \). This tells us that for every unit the line moves horizontally, it rises by 3 units.

Equation of a Line

The equation of a line is a mathematical expression that describes all the points along a line. In general, equations of a line can be written in various forms, each providing a different perspective. For instance, the slope-intercept form shows the slope and the y-intercept directly, which is why it is widely used. Another common form is the standard form, which looks like \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. In our example, the original equation is \( 3x - y = 0 \). This is already in the standard form, making it easy to work with when converting to other forms and finding special line properties like perpendicularity.

Point-Slope Form

The point-slope form of a line is especially useful when you have a specific point on a line and the slope, and you need to find the equation of the line. It is given by the formula \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This form is ideal for problems like this one, where we know a point the line passes through (given as \( (1, 3) \)) and the slope \( m = -\frac{1}{3} \), which is the negative reciprocal of the original line, indicating perpendicularity. By substituting these values in, we can quickly find the new line's equation in a straightforward way.

Standard Form of a Line

The standard form of a line \( Ax + By = C \) provides a clean and organized way to express a line equation, especially useful for calculating intercepts and dealing with integer coefficients. Although converting from other forms to standard form can sometimes require careful algebraic manipulation, it often simplifies certain types of problems. In our problem, the line \( 3x - y = 0 \) was initially in standard form. Later, the point-slope equation \( y - 3 = -\frac{1}{3}(x - 1) \) transforms into \( y = -\frac{1}{3}x + \frac{10}{3} \), highlighting how standard form can initially present an equation before simplifying or reformatting it for a specific purpose.