/*! 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 53 Write an equation of the line in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 53

Write an equation of the line in slope-intercept form, if possible, given the slope and a point that lies on. Slope: \(m=-3\) (-2,2)

Short Answer

Expert verified
The equation of the line is \(y = -3x - 4\).

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 \(m\) is the slope and \(b\) is the y-intercept. We are given that the slope \(m\) is \(-3\) and a point that the line passes through is \((-2, 2)\).
02

Substitute the Slope into the Equation

Substitute the given slope \(m = -3\) into the slope-intercept form equation. This gives us \(y = -3x + b\).
03

Use the Given Point to Solve for b

Substitute the point \((-2, 2)\) into the equation \(y = -3x + b\) to find \(b\). Replace \(x\) with \(-2\) and \(y\) with \(2\): \[2 = -3(-2) + b\].
04

Solve for b

Calculate the equation from the substitution: \[2 = 6 + b\]. Solving for \(b\) gives \(b = 2 - 6\), which simplifies to \(b = -4\).
05

Write the Final Equation in Slope-Intercept Form

Now that we know \(m = -3\) and \(b = -4\), substitute these back into the slope-intercept equation to get the final equation of the line: \(y = -3x - 4\).

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.

Linear Equations
Linear equations are fundamental in algebra and are represented in the form of a straight line when graphed on a coordinate plane. These equations often describe a line鈥檚 relationship between two variables, usually represented as "x" and "y". The most common form of a linear equation is the slope-intercept form, which is an easy way to understand the characteristics of the line.
  • Slope-Intercept Form: This format is expressed as \( y = mx + b \), where "m" denotes the slope of the line, and "b" denotes the y-intercept.
  • Why Use Linear Equations? Linear equations are useful to model real-world situations that involve constant change and are an essential tool in understanding the patterns between two variables.
Linear equations are straightforward to solve, especially when you know the values of the slope and the y-intercept. They offer us a simple way to predict and calculate results for given values of "x".
Slope
The slope is a measure of the steepness or incline of a line, represented by the letter "m" in the slope-intercept form \( y = mx + b \). It is a crucial component in determining the direction and angle of the line when graphed.
  • Positive vs. Negative Slope: A positive slope means the line ascends from left to right, indicating a direct relationship between the variables. Conversely, a negative slope, like \( m = -3 \) in our example, means the line descends from left to right, showcasing an inverse relationship.
  • Interpreting the Value: The numeric value of the slope tells us how much "y" changes for a unit change in "x". For example, a slope of \(-3\) implies that for every one unit increase in "x", "y" decreases by 3 units.
Understanding the slope helps in predicting how changes in one variable will affect the other, making it a fundamental concept in analyzing and interpreting linear relationships.
Y-Intercept
The y-intercept is a key feature of linear equations in the slope-intercept form. It appears as the value "b" in \( y = mx + b \). This point on a graph indicates where the line crosses the y-axis, providing a starting point for the line.
  • Importance of the Y-Intercept: The y-intercept directly shows us the value of "y" when "x" equals zero. This means it is the initial value when there is no influence from the independent variable "x".
  • Real-World Applications: In practical scenarios, the y-intercept can represent things like starting fees, fixed costs, or baseline quantities. For example, if a taxi company charges a flat rate before the journey starts, this would be analogous to the y-intercept in the equation.
In our example, the y-intercept was calculated to be \(-4\), saying that when "x" is zero, the value of "y" is \(-4\). Having this point allows us to more accurately graph and interpret the line.

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

Refer to the following: From March 2000 to March 2008, data for retail gasoline price in dollars per gallon are given in the table below. (These data are from Energy Information Administration, Official Energy Statistics from the U.S. Government at http://tonto.eia.doe.gov/oog/info/gdu/gaspump.html.) Use the calculator [STAT] [EDIT] command to enter the table below with \(L_{1}\) as the year (x 1 for year 2000) and \(L_{2}\) as the gasoline price in dollars per gallon. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|}\hline \text { March of Each vear } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\\\\hline \text { Rerale casoune Price } \$ \text { PER cautuon } & 1.517 & 1.409 & 1.249 & 1.693 & 1.736 & 2.079 & 2.425 & 2.563 & 3.244 \\\\\hline\end{array}$$ a. Use the calculator commands [STAT] [LinReg]to model the data using the least squares regression. Find the equation of the least squares regression line using \(x\) as the year \((x=1 \text { for year } 2000)\) and \(y\) as the gasoline price in dollars per gallon. Round all answers to three decimal places. b. Use the equation to determine the gasoline price in March \(2006 .\) Round all answers to three decimal places. Is the answer close to the actual price? c. Use the equation to find the gasoline price in March \(2009 .\) Round all answers to three decimal places.

Determine whether each statement is true or false. If a line has no slope (undefined slope), describe a line that is parallel to it.

The value of a Daewoo car is given by \(y=11,100-1850 x, x \geq 0,\) where \(y\) is the value of the car and \(x\) is the age of the car in years. Find the \(x\) -intercept and \(y\) -intercept and interpret the meaning of each.

Determine whether the lines are parallel, perpendicular, or neither, and then graph both lines in the same viewing screen using a graphing utility to confirm your answer. $$\begin{aligned} &y_{1}=\frac{1}{2} x+5\\\ &y_{2}=2 x-3 \end{aligned}$$

Refer to the following: Einstein's special theory of relativity states that time is relative: Time speeds up or slows down, depending on how fast one object is moving with respect to another. For example, a space probe traveling at a velocity \(v\) near the speed of light \(c\) will have "clocked" a time \(t\) hours, but for a stationary observer on Earth that corresponds to a time \(t_{0} .\) The formula governing this relativity is given by $$ t=t_{0} \sqrt{1-\frac{v^{2}}{c^{2}}} $$ If the time elapsed on a space probe mission is 18 years but the time elapsed on Earth during that mission is 30 years, how fast is the space probe traveling? Give your answer relative to the speed of light.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.