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Chapter 3: Problem 55

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-\frac{4}{7},\) through (-1,-2)

Short Answer

Expert verified
y = -\frac{4}{7}x - \frac{18}{7}

Step by step solution

01

Understand the Slope-Intercept Form

The equation of a line in slope-intercept form is given by \( y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept. We have a slope \( m = -\frac{4}{7} \).
02

Use the Point-Slope Formula

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Use the given point \( (-1, -2) \) and slope \( -\frac{4}{7} \): \( y - (-2) = -\frac{4}{7}(x - (-1)) \).
03

Simplify the Equation

Simplify \( y + 2 = -\frac{4}{7}(x + 1) \). Distribute \( -\frac{4}{7} \): \( y + 2 = -\frac{4}{7}x - \frac{4}{7} \).
04

Solve for y in Slope-Intercept Form

Isolate \( y \) by subtracting \( 2 \) from both sides: \( y = -\frac{4}{7}x - \frac{4}{7} - 2 \). Convert \( 2 \) to \( \frac{14}{7} \): \( y = -\frac{4}{7}x - \frac{4}{7} - \frac{14}{7} \). Combine the constants: \( y = -\frac{4}{7}x - \frac{18}{7} \).

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Key Concepts

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

Equation of a Line
An equation of a line represents a straight path that continues infinitely in both directions on a graph. To form this equation, it's important to know a point on the line and the slope, or how steep the line is. Lines can be expressed in multiple forms, but a common one is the slope-intercept form, which we'll explore further.
  • The equation of a line conveys its slope and intercept;
  • These equations can be rearranged based on what information is given;
  • Understanding different forms allows for flexibility when solving or graphing equations.
Recognizing the parts of the equation and how they interact is key to mastering linear equations in analytics and graphing settings.
Point-Slope Form
The point-slope form is another way to write an equation of a line, especially useful when a point on the line and the slope are known. Unlike the slope-intercept form, which highlights the y-intercept, this form uses:
  • A known point \( (x_1, y_1) \) on the line;
  • The slope, \( m \), of the line.
The equation is expressed as \( y - y_1 = m(x - x_1) \). This form is ideal for quickly forming an equation with inputted values. For example, using the point \( (-1, -2) \) and slope \( -\frac{4}{7} \), we can substitute these values directly into the formula, making this form convenient for transformations into other formats like slope-intercept.
Slope and Intercept
Understanding slope and intercept is fundamental when working with linear equations. The slope \( m \) measures the steepness of a line, indicating how much the y-value changes with each step along the x-axis.
  • The slope is positive if the line rises as it moves right;
  • It's negative if the line falls as it moves right.
The intercept, on the other hand, is where the line crosses the y-axis, noted by the 'c' in the slope-intercept form \( y = mx + c \). Thus, every line can be defined by these two characteristics. In our exercise, the line's slope \( -\frac{4}{7} \) tells us it decreases as x increases, showing it's slanted downward with the y-intercept at \( -\frac{18}{7} \).
Linear Equations
Linear equations form the backbone of many mathematical concepts, representing constant rates of change and models of proportionality. These equations, like the one \( y = -\frac{4}{7}x - \frac{18}{7} \), produce straight lines when graphed and analyze the relationship between two variables.
  • They follow the pattern of a first-degree polynomial;
  • All their points align perfectly along one straight line.
The simplicity of linear equations makes them incredibly useful not only in mathematics but also in physics, economics, and engineering to illustrate trends and make predictions based on data sets. The ease of transforming one form to another only adds to their versatility and application.

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Most popular questions from this chapter

Solve. See the Concept Checks in this section. Is the graph of (3,0) in the same location as the graph of (0,3) ? Explain why or why not.

Solve. See Example 4. The table shows the domestic box office (in billions of dollars) for the U.S. movie industry during the years shown. (Source: Motion Picture Association of America) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Box Office (in billions of dollars) } \\ \hline 2003 & 9.17 \\ \hline 2004 & 9.22 \\ \hline 2005 & 8.83 \\ \hline 2006 & 9.14 \\ \hline 2007 & 9.63 \\ \hline 2008 & 9.79 \\ \hline \end{array} $$ a. Write this paired data as a set of ordered pairs of the form (year, box office). b. In your own words, write the meaning of the ordered pair (2006,9.14) c. Create a scatter diagram of the paired data. Be sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

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Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2004,\) there were approximately 83,000 gaselectric hybrid vehicles sold in the United States. In \(2007,\) there were approximately 353,000 such vehicles sold. (Source: Energy Information Administration, Department of Energy) a. Assume the relationship between years past 2004 and the number of vehicles sold is linear over this period. Write an equation describing the relationship between time and the number of gas-electric hybrid vehicles sold. Use ordered pairs of the form (years past \(2004,\) number of vehicles sold). b. Use this equation to estimate the number of gas-electric hybrid sales in \(2009 .\)
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