Problem 32 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-1,-1)\) and perpendicular to $$ x-y+3=0 $$

Step-by-Step Solution

Answer

The standard form of the line is \(x + y = -2\).

Step 1: Identify the Slope of the Given Line

The standard form of the given line is \(x - y + 3 = 0\). To identify the slope, convert this equation into slope-intercept form. This gives \(y = x + 3\). The slope \(m\) of this line is \(1\).

Step 2: Determine the Perpendicular Slope

The slope of the line perpendicular to the given line will be the negative reciprocal of \(1\). Therefore, the perpendicular slope is \(-1\).

Step 3: Use Point-Slope Form

Utilize the point-slope form: \(y - y_1 = m(x - x_1)\), where \(m = -1\) and the point \((-1, -1)\). Substitute these values in to get: \(y + 1 = -1(x + 1)\).

Step 4: Simplify to Slope-Intercept Form

Simplify the equation from Step 3: \(y + 1 = -1x - 1\). This simplifies to \(y = -x - 2\).

Step 5: Convert to Standard Form

To put \(y = -x - 2\) into standard form, rearrange to get \(x + y = -2\). This is the standard form \(Ax + By = C\), where \(A = 1\), \(B = 1\), and \(C = -2\).

Key Concepts

Standard Form Equation

In coordinate geometry, the standard form equation of a line is represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. This format is especially useful because it neatly expresses the relationship between \( x \) and \( y \) in a linear equation. One of its great advantages is that it makes it easy to quickly identify key characteristics of the line, such as its intercepts.

• To convert from slope-intercept form to standard form, rearrange the terms so that \( x \) and \( y \) are on one side and the constant \( C \) is on the other.
• Ensure that \( A \), \( B \), and \( C \) are integers and \( A \) is non-negative for consistency.

Standard form is often preferred for geometric calculations on graphs because it provides a more intuitive grasp of where the line crosses the axes.

Slope-Intercept Form

The slope-intercept form is one of the most familiar ways to express a line equation. Written as \( y = mx + b \), it has two key components:

• \( m \) is the slope, indicating the steepness or incline of the line. It shows the change in \( y \) for every unit change in \( x \).
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

This form is particularly handy for quickly assessing how the line behaves with respect to the y-axis. If given a line not in this form, converting it can offer insights into the line's slope and intercept, allowing for easier graphing and comparison with other lines. To switch from this format to standard form, you simply reorganize the terms to align with the standard form structure, ensuring that the coefficients are integers.

Negative Reciprocal

The concept of a negative reciprocal is crucial when dealing with perpendicular lines in geometry. If you have a line with a slope \( m \), a line that is perpendicular to it will have a slope that is the negative reciprocal of \( m \).

• The negative reciprocal of a slope \( m \) can be found by taking \(-1/m\).
• This method ensures that two lines intersect to form a right angle (90 degrees).

For instance, if a line has a slope of \( 1 \), the slope of a line perpendicular to it would be \(-1\). Applying this concept is straightforward as it often involves flipping and changing the sign of the original slope. Understanding these relationships between slopes allows you to analyze and predict the behavior of lines on a graph, ensuring precision in geometric interpretations.