Problem 43 - 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 \(P\). \(l: y=2 x+1 ; P=(-1,-3)\)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{2}x - \frac{7}{2} \).

Step 1: Identify the Slope of Line l

The given line has the equation \( y = 2x + 1 \). The slope of this line is 2, as it is the coefficient of \( x \) in the equation of the line.

Step 2: Find the Perpendicular Slope

To find the slope of a line that is perpendicular to the given line, we take the negative reciprocal of the slope of line \( l \). Hence, the perpendicular slope is \( -\frac{1}{2} \).

Step 3: Use Point-Slope Form

The point-slope form of a line is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Substitute \( m = -\frac{1}{2} \) and \( P(-1, -3) \) into the formula to get:\[ y - (-3) = -\frac{1}{2}(x - (-1)) \] Which simplifies to: \[ y + 3 = -\frac{1}{2}(x + 1) \]

Step 4: Simplify the Equation

Distribute the \( -\frac{1}{2} \) on the right side to get: \[ y + 3 = -\frac{1}{2}x - \frac{1}{2} \]. Subtract 3 from both sides to isolate \( y \): \[ y = -\frac{1}{2}x - \frac{1}{2} - 3 \]. Further simplify: \[ y = -\frac{1}{2}x - \frac{7}{2} \].

Key Concepts

Slope of a Line

The slope of a line is a key concept in understanding linear equations. It describes how steep a line is, and is often represented by the letter \(m\). In an equation of the line in slope-intercept form \(y = mx + b\), the slope is the coefficient of \(x\). For the line \(y = 2x + 1\), the slope \(m\) is 2.

To find perpendicular lines, we use the negative reciprocal of the slope of the original line. This means that if the slope of a line is \(a\), the slope of a line perpendicular to it will be \(-\frac{1}{a}\). So for our line, with slope 2, the slope of a line perpendicular would be \(-\frac{1}{2}\). When two lines are perpendicular, their slopes multiply to give \(-1\). This is a handy check to ensure that you've found the right slope for the perpendicular line.

Point-Slope Form

The point-slope form is a way to write linear equations using the slope of the line and a specific point through which the line passes. It is especially useful when you know one point on a line and the slope, but not the full equation yet.

The formula for the point-slope form of a line is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the known point on the line, and \(m\) is the slope. For example, let's say we want the equation of a line passing through \((-1, -3)\) with a slope of \(-\frac{1}{2}\).

Substitute these values into the point-slope formula:
\[ y - (-3) = -\frac{1}{2}(x - (-1)) \]
Which simplifies to:
\[ y + 3 = -\frac{1}{2}(x + 1) \]
This sets up the equation perfectly to transition into a format that might be more desirable for certain applications, such as the slope-intercept form, or to just see the general behavior of the line more clearly.

Equation of a Line

Once we have used the point-slope form, we can rearrange terms to express the equation in a more general form. This form is often the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

By starting from the point-slope form \(y + 3 = -\frac{1}{2}(x + 1)\), we distribute the \(-\frac{1}{2}\) on the right side:
\[ y + 3 = -\frac{1}{2}x - \frac{1}{2} \]
Next, we isolate \(y\) by subtracting 3 from both sides:
\[ y = -\frac{1}{2}x - \frac{1}{2} - 3 \]
Simplifying further, we combine these terms to get:
\[ y = -\frac{1}{2}x - \frac{7}{2} \]
This is the equation of the line that is perpendicular to the original line \(l\) that passes through the point \((-1, -3)\). This final equation tells us everything about the line's slope and where it crosses the y-axis, simplifying several types of geometric analysis.