Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((5,-9)\) and perpendicular to the line whose equation is \(x+7 y=12\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \( y + 9 = 7(x - 5) \), and in slope-intercept form is \( y = 7x - 44 \).

Step 1: Find the slope of the given line

The equation of the given line is \(x+7y=12\). Rewrite this in slope-intercept form (i.e. \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept), by isolating \(y\). \[ 7y = -x + 12 \] Which simplifies to: \[ y = -\frac{1}{7}x + \frac{12}{7} \] So, the slope (\(m_1\)) of the given line is -1/7.

Step 2: Find the slope of the perpendicular line

Recall that the slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Thus, the slope (\(m_2\)) of the line perpendicular to the given line is the negative reciprocal of -1/7, which is 7.

Step 3: Write the equation of the line in point-slope form

The point-slope form of a line passing through the point \((x_1, y_1) = (5,-9)\) with slope \(m_2 = 7\) is: \[ y - y_1 = m_2(x - x_1) \] Substituting \(x_1, y_1, m_2\) with 5, -9, 7 respectively gives: \[ y - (-9) = 7(x - 5) \] which simplifies to: \[ y + 9 = 7x - 35 \]

Step 4: Convert the equation into slope-intercept form

Rewrite the equation in slope-intercept form by solving for \(y\) to get: \[ y = 7x - 44 \]

Key Concepts

Point-Slope Form

The point-slope form is a fundamental concept in understanding the equation of a line. It is particularly useful when you know a point on the line and its slope.
A line can be expressed in the point-slope form as:

• \[ y - y_1 = m(x - x_1) \]

Here,

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

This form is handy for constructing lines when new conditions are given. Using the point-slope formula, you can quickly formulate the equation of a line without needing the y-intercept at the start. For example, if you have a slope of 7 and the point (5, -9), you simply substitute them into the formula: \( y + 9 = 7(x - 5) \).
This format allows for straightforward writing and conversion of line equations, especially in problems involving perpendicular or parallel lines. Understanding point-slope form brings you a step closer to mastering all the other forms.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used forms to represent a linear equation. It is popular due to its simplicity and the immediate information it provides about a line.
The general form of a line in slope-intercept form is:

• \[ y = mx + b \]

Where:

• \( m \) is the slope of the line
• \( b \) is the y-coordinate of the y-intercept, where the line crosses the y-axis

When you need to convert an equation into the slope-intercept form, you solve for \( y \). In the exercise, after creating the line's equation using point-slope form, it was transformed into slope-intercept form: \( y = 7x - 44 \).
This form simplifies comparing line equations or graphing them, as it directly tells you two crucial line attributes: slope and y-intercept. It's a highly efficient method for determining line properties at a glance.

Equation of a Line

The equation of a line is a mathematical statement that descripes all the points on a line. Understanding line equations can be achieved by exploring various forms, where each form offers a different perspective or utility based on the information available or needed.
We delve into two popular forms: the point-slope form and the slope-intercept form. These forms help you understand and articulate the properties and positions of lines efficiently. Formulating equations of lines involves manipulating these forms based on given conditions.
For instance, if a line is perpendicular to another, you use the negative reciprocal of the original line's slope to find the new slope. When creating the equation of a line that is perpendicular to another, as in our exercise, first rewrite the given line in slope-intercept form. Then, determine the perpendicular line's slope, and proceed to create the new line's equation using the point-slope form. Finally, convert it into the slope-intercept form for clarity and simplicity. Mastering these steps allows for a deep understanding of linear relationships, enhancing your problem-solving skills in geometry and algebra.