Textbook Question
Solve: .
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 85
Write an equation in point-slope form and slope-intercept form for the line passing through (-2, -6) and perpendicular to the line whose equation is x − 3y+ 9 = 0.
Verified step by step guidance1
First, rewrite the given line equation \(x - 3y + 9 = 0\) in slope-intercept form \(y = mx + b\) to identify its slope. Start by isolating \(y\): \(x - 3y + 9 = 0\) becomes \(-3y = -x - 9\), then divide both sides by \(-3\) to get \(y = \frac{1}{3}x + 3\).
From the slope-intercept form, identify the slope of the given line as \(m = \frac{1}{3}\). Since the line we want is perpendicular to this line, find the perpendicular slope by taking the negative reciprocal: \(m_{\perp} = -3\).
Use the point-slope form formula \(y - y_1 = m(x - x_1)\) with the point \((-2, -6)\) and the perpendicular slope \(m_{\perp} = -3\). Substitute to get \(y - (-6) = -3(x - (-2))\), which simplifies to \(y + 6 = -3(x + 2)\).
To write the equation in slope-intercept form, expand the right side: \(y + 6 = -3x - 6\). Then isolate \(y\) by subtracting 6 from both sides: \(y = -3x - 12\).
Now you have both forms: the point-slope form \(y + 6 = -3(x + 2)\) and the slope-intercept form \(y = -3x - 12\) for the line passing through \((-2, -6)\) and perpendicular to the given line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope of a Line
The slope measures the steepness of a line and is calculated as the ratio of the change in y to the change in x between two points. For a line in standard form Ax + By + C = 0, the slope is -A/B. Understanding slope is essential for finding the slope of the given line and its perpendicular counterpart.
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The Slope of a Line
Perpendicular Lines and Their Slopes
Two lines are perpendicular if the product of their slopes is -1. This means the slope of a line perpendicular to another is the negative reciprocal of the original line's slope. This concept helps determine the slope of the line required in the problem.
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Point-Slope and Slope-Intercept Forms of a Line
Point-slope form is written as y - y₁ = m(x - x₁), using a point (x₁, y₁) and slope m. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Both forms are used to express the equation of a line once the slope and a point are known.
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