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Chapter 3: Problem 10

Write an equation of the line with each given slope, \(m,\) and \(y\) -intercept, \((0, b) .\) See Example \(1\). $$ m=\frac{1}{2}, b=-\frac{1}{3} $$

Short Answer

Expert verified
The equation is \( y = \frac{1}{2}x - \frac{1}{3} \).

Step by step solution

01

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \). In this formula, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
02

Substitute Given Values

Now that we know the slope \( m = \frac{1}{2} \) and the y-intercept \( b = -\frac{1}{3} \), we can substitute these values into the slope-intercept equation \( y = mx + b \).
03

Write the Equation

Replace \( m \) with \( \frac{1}{2} \) and \( b \) with \( -\frac{1}{3} \) in the equation. This gives us: \[ y = \frac{1}{2}x - \frac{1}{3} \]. This is the equation of the line with the given slope and y-intercept.

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Key Concepts

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

Linear Equation
A linear equation is a fundamental concept in algebra. It's called "linear" because its graph is a straight line. The standard form of a linear equation is expressed in the format:
  • Slope-Intercept Form: \( y = mx + b \)
In this equation:
  • \( y \) represents the dependent variable or output.
  • \( x \) is the independent variable or input.
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept.
Every point on a straight line solution of a linear equation satisfies this equation when substituted for \( x \) and \( y \). This makes it very useful for predicting outcomes or describing relationships between quantities.
Linear equations can model various real-world situations, such as predicting costs, distances, and other continuous data.
Slope
The slope of a line is a measure of how steep the line is. It's a key component in the slope-intercept form of a linear equation. The slope can be thought of as the "rise over run," or more formally, the change in the y-value divided by the change in the x-value. Mathematically, it is represented as:
  • \( m = \frac{\text{change in } y}{\text{change in } x} \)
In simpler terms, the slope tells us:
  • Horizontal Lines: A slope of 0 indicates a horizontal line, meaning there is no vertical change as you move along the line.
  • Vertical Lines: An undefined slope is associated with a vertical line, where there is no horizontal movement.
The slope can be positive or negative:
  • Positive slope: The line rises as it moves from left to right.
  • Negative slope: The line falls as it moves from left to right.
In our exercise, the slope \( m = \frac{1}{2} \) means that for every 2 units we move horizontally, the line rises 1 unit. Understanding the slope helps in grasping how certain changes in input affect the output.
Y-Intercept
The y-intercept is another critical aspect of understanding linear equations. It is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the variable \( b \) is the y-intercept.
The y-intercept provides insight into the starting point or initial condition of a line when \( x = 0 \). For any linear equation:
  • The y-intercept is found by setting \( x \) to zero in the equation and solving for \( y \).
  • This intercept informs us where the line will intersect the vertical axis on a graph.
In the context of our exercise, the y-intercept is \( b = -\frac{1}{3} \), implying that the line intersects the y-axis at \( -\frac{1}{3} \). This means that when no other variables are affecting the equation (when \( x = 0 \)), the value of \( y \) would be \(-\frac{1}{3}\). Understanding the y-intercept allows one to quickly determine the initial state of a relationship between the variables involved in the linear equation.

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Most popular questions from this chapter

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In January 2007 , there were \(71,000\) registered gasolineelectric hybrid cars in the United States. In \(2004,\) there were only \(29,000\) registered gasoline- electric hybrids. (Source: U.S. Energy Information Administration) a. Write an equation describing the relationship between time and number of registered gasoline-hybrid cars. Use ordered pairs of the form (years past \(2004,\) number of cars). b. Use this equation to predict the number of gasolineelectric hybrids in the year 2010 .

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. \((-3,-3)\) and \((0,0)\)

Find the slope of the line through the given points. \((2.3,0.2)\) and \((7.9,5.1)\)

Write the equations of three lines parallel to \(10 x-5 y=-7\) (See the third Concept Check.)

Solve. See the Concept Check in this section. In your own words, describe how to plot or graph an ordered pair of numbers.
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