/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Write an equation of each line w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 8

Write an equation of each line with the given slope and containing the given point. Write the equation in the slope-intercept form \(y=m x+b .\) See Example \(1 .\) Slope \(-\frac{1}{5} ;\) through (4,-6)

Short Answer

Expert verified
The equation of the line is \( y = -\frac{1}{5}x - \frac{26}{5} \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept, which is where the line crosses the y-axis. We need to find both \( m \) and \( b \) to write the equation.
02

Substitute the Slope

We are given a slope of \( -\frac{1}{5} \). Substitute this into the equation \( y = mx + b \) to get \( y = -\frac{1}{5}x + b \). This will help us in finding the y-intercept \( b \).
03

Use the Point to Solve for b

We need to substitute the given point \((4, -6)\) into the equation \( y = -\frac{1}{5}x + b \). Using \( x = 4 \) and \( y = -6 \), we substitute them in:\[ -6 = -\frac{1}{5}(4) + b \]Simplify the equation:\[ -6 = -\frac{4}{5} + b \]
04

Solve for the Y-Intercept (b)

Continuing from the previous step:Add \( \frac{4}{5} \) to both sides of the equation to solve for \( b \):\[ -6 + \frac{4}{5} = b \]Convert \( -6 \) to a fraction: \( -\frac{30}{5} \)\[ -\frac{30}{5} + \frac{4}{5} = b \]\[ b = -\frac{26}{5} \]
05

Write the Final Equation

Now that we have both \( m \) and \( b \), substitute them back into the slope-intercept form \( y = mx + b \):\[ y = -\frac{1}{5}x - \frac{26}{5} \]This is the equation of the line with the given slope that goes through the point \((4, -6)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line
The equation of a line is a powerful tool in algebra that represents a straight line on a coordinate plane. In its simplest form, the equation is expressed as \( y = mx + b \). The two main components of this equation are the slope \( m \) and the y-intercept \( b \). Understanding these components can help in visualizing how the line behaves. The slope \( m \) indicates the steepness and direction of the line, while the y-intercept \( b \) shows where the line crosses the y-axis. By knowing these, we can predict and calculate every point along the line. A line鈥檚 equation consolidates an infinite number of points into a concise algebraic expression.
Finding Y-Intercept
Finding the y-intercept \( b \) is crucial when writing the equation of a line in slope-intercept form. The y-intercept is the point where the line crosses the y-axis, meaning it occurs at \( x = 0 \). To find \( b \), you begin with substituting the known values into the equation \( y = mx + b \). If the slope \( m \) and a point \((x, y)\) are known, simply rearrange and solve for \( b \).
  • Insert the values of the slope and the coordinates into the equation.
  • Simplify the resulting expression by solving for \( b \).
This isolates the y-intercept, allowing you to determine exactly where the line will intersect the y-axis.
Using Given Point for Equations
When given a specific point on the line, such as \((4, -6)\), this information can be used to complete the equation of the line. Inserting the x and y coordinates of the point into the equation \( y = mx + b \) assists in solving for the y-intercept \( b \). The process is straightforward:
  • Begin with the equation with the known slope and unknown \( b \), such as \( y = -\frac{1}{5}x + b \).
  • Substitute the x-value and y-value from the point into the equation.
  • Rearrange the equation to solve for \( b \).
Understanding how to integrate a point into the equation ensures accuracy and helps verify the line passes through this specific point.
Algebraic Problem Solving
Algebraic problem solving involves using algebraic methods to find unknown values. In this context, solving for \( b \) requires manipulating the equation \( y = mx + b \) to isolate \( b \). This involves combining like terms and adjusting both sides of the equation until you have \( b = \) (some value). For example:
  • Follow the arithmetic operations to simplify the expressions.
  • Convert whole numbers to fractions if needed, to match denominators.
  • Ensure all calculations are properly executed, especially when fractions are involved.
These problem-solving skills are widely applicable in mathematics, helping you to solve not only simple line equations but also more complex algebraic expressions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ g(-6) $$

For each function, find the indicated values. \(f(x)=\frac{1}{2} x ;\) a. \(f(0)\) b. \(f(2)\) c. \(f(-2)\)

The per capita consumption (in pounds) of all beef in the United States is given by the function \(C(x)=-0.33 x+67.1,\) where \(x\) is the number of years since \(2000 .\) (Source: Based on data from the Economic Research Service, U.S. Department of Agriculture, \(2000-2008\) ) a. Find and interpret \(C(4)\). b. Estimate the per capita consumption of beef in the United States in \(2010 .\)

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7,\) find each function. $$ Q(x)-R(x) $$

For each function, find the indicated values. \(f(x)=2 x+7\) a. \(f(2)\) b. \(f(a)\) c. \(f(-x)\) d. \(f(x+h)\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.