Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) \text { Through }(2,2) \text { with slope }-1

Step-by-Step Solution

Answer

The equation of the line is \(x + y - 4 = 0\).

Step 1: Understand the problem

We need to write an equation for a line that passes through a specific point, (2, 2), and has a specified slope, which is -1. The final equation should be in the form \(Ax + By + C = 0\).

Step 2: Use the point-slope form of a line

The point-slope form of a line is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Here, the slope \(m = -1\) and the point is \((2, 2)\). Substitute these values into the equation: \(y - 2 = -1(x - 2)\).

Step 3: Simplify the equation

Simplify the equation from Step 2: \(y - 2 = -x + 2\). Add 2 to both sides to obtain \(y = -x + 4\).

Step 4: Rearrange to the standard form

We need to convert the equation \(y = -x + 4\) into the form \(Ax + By + C = 0\). Start by moving all terms to one side: \(x + y - 4 = 0\), which is equivalent to \(1 \cdot x + 1 \cdot y - 4 = 0\).

Step 5: Write the final equation

The equation of the line in the form \(Ax + By + C = 0\) is \(x + y - 4 = 0\).

Key Concepts

Point-slope form

The point-slope form of a line is a straightforward method to write the equation of a line when you know one point on the line and the line's slope. This format prevents any confusion and ensures accuracy in plotting or graphing the line. The formula is expressed as \( y - y_1 = m(x - x_1) \). Here, \( m \) represents the slope of the line, and the coordinates \( (x_1, y_1) \) are the specific point the line passes through. For example, when given the point (2, 2) and a slope of -1, you would substitute these values into the point-slope form equation to get:

• Step 1: Substitute \( y - 2 = -1(x - 2) \).
• Step 2: Distribute the slope: \( y - 2 = -x + 2 \).

This provides a practical representation of the line, showing the relationship between the slope and the position of the point used.

Slope-intercept form

The slope-intercept form makes it very easy to graph a line, as it directly reveals the slope and the y-intercept. The equation follows the format \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept—where the line crosses the y-axis. After simplifying the point-slope equation, you arrive at this form. From our earlier example with \( y - 2 = -x + 2 \), after rearranging as \( y = -x + 4 \), the slope-intercept form becomes clear:

• The slope \( m \) is -1.
• The y-intercept \( b \) is 4.

This means the line crosses the y-axis at 4 and moves down one unit vertically for each unit it moves right horizontally. It is an intuitive form that illustrates both the steepness and the starting point of the line.

Standard form of a line

The standard form of a line is a traditional way to express linear equations. This form is given by \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should ideally be a non-negative integer. This form is beneficial because it emphasizes both variables on one side of the equation. Starting from the slope-intercept form of \( y = -x + 4 \), you can rearrange it to fit the standard form by moving all terms to one side:

• Step 1: Add \( x \) to both sides: \( x + y = 4 \).
• Step 2: Subtract 4 to get: \( x + y - 4 = 0 \).

This simplifies to \( 1 \cdot x + 1 \cdot y - 4 = 0 \), which is the line's equation in standard form. This format is often used in algebra because it standardizes linear equations for further analysis or operations, like solving systems of equations.