Problem 30 - 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 \(A x+B y+C=0\). Through \((3,4)\) with slope \(-1\)

Step-by-Step Solution

Answer

The equation is \(x + y - 7 = 0\).

Step 1: Identify the Point-Slope Form

We are given a point \((3, 4)\) and a slope \(m = -1\). The point-slope form of a line's equation is given by:\[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is the given point. Substitute \((x_1, y_1) = (3, 4)\) and \(m = -1\) into this equation.

Step 2: Substitute into Point-Slope Form

Substitute the values into the point-slope equation:\[ y - 4 = -1(x - 3) \]This is the equation of the line in point-slope form.

Step 3: Simplify the Point-Slope Equation

Expand and simplify the equation:\[ y - 4 = -x + 3 \]Add 4 to both sides to isolate \(y\) on one side:\[ y = -x + 7 \]

Step 4: Convert to Standard Form

Rewrite the equation \(y = -x + 7\) to standard form \(A x + B y + C = 0\) by rearranging terms:\[ x + y - 7 = 0 \]Here, \(A = 1\), \(B = 1\), and \(C = -7\). This is the standard form of the line's equation.

Key Concepts

point-slope form

The point-slope form is a way to express the equation of a straight line. This form is particularly useful when you have a specific point on the line and the slope of the line itself. The equation is written as:\[ y - y_1 = m(x - x_1) \]Here:

• \((x_1, y_1)\) represents a known point on the line.
• \(m\) is the slope of the line.

This formula is handy because it directly uses a point and a slope, making it straightforward to write an equation for any line. To use the point-slope form, simply plug in the coordinates of the point and the slope into the formula. For example, if the point is \(3, 4\) and the slope is \(-1\), the equation becomes:\[ y - 4 = -1(x - 3) \]Simplifying from here can lead directly to other forms of a linear equation, like the slope-intercept form.

linear equations

Linear equations are a fundamental concept in algebra. These equations represent straight lines when graphed on the coordinate plane. A typical linear equation can be written in several forms, with each form offering specific advantages depending on the information given.Key characteristics of linear equations include:

• The highest power of the variable(s) is one.
• The graph of a linear equation is always a straight line.
• A linear equation in two variables, \(x\) and \(y\), can be written as \(Ax + By + C = 0\).

Analyzing linear equations involves understanding the relationship between the slope and intercepts, and converting the equation to a more convenient form like the point-slope or slope-intercept form can often simplify calculations and interpretations. These conversions make linear equations versatile tools for solving real-world problems.

slope-intercept form

The slope-intercept form provides an intuitive way to express a linear equation, clearly showing both the slope of the line and where it intersects the y-axis. The formula for slope-intercept form is given by:\[ y = mx + b \]Here:

• \(m\) is the slope of the line, indicating the line's steepness and direction.
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

This form is especially popular due to its simplicity and ease of use when graphing a line. From the slope-intercept form, you can easily determine how to draw the line on a graph. For example, from the equation \(y = -x + 7\), it's clear that the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(7\), meaning the line crosses the y-axis at \(7\) and trends downward as it moves from left to right.