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Chapter 1: Problem 7

Change the equation \(3=2 y\) to the form \(y=m x+b\), graph the line, and find the \(y\) -intercept and \(x\) -intercept. \(\Rightarrow\)

Short Answer

Expert verified
The equation is \(y = 0x + \frac{3}{2}\); the \(y\)-intercept is \((0, \frac{3}{2})\), and there is no \(x\)-intercept.

Step by step solution

01

Understanding the Equation

The equation given is \(3 = 2y\). This is currently in a form not compatible with \(y = mx + b\). We need to manipulate it to be in terms of \(y\) on the left side.
02

Solving for y

We rearrange the equation \(3 = 2y\) to solve for \(y\). Divide both sides by 2 to isolate \(y\): \[y = \frac{3}{2}\].
03

Converting to y = mx + b Form

Since the equation \(y = \frac{3}{2}\) does not include \(x\), it means \(m = 0\), and therefore the equation is \(y = 0\cdot x + \frac{3}{2}\). This is in the form \(y = mx + b\), where \(m = 0\) and \(b = \frac{3}{2}\).
04

Graphing the Line

On the graph, the line is horizontal because the slope \(m = 0\). It intersects the \(y\)-axis at \(\frac{3}{2}\), meaning it is a line parallel to the \(x\)-axis at \(y = \frac{3}{2}\).
05

Finding y-intercept

The equation \(y = mx + b\) directly gives the \(y\)-intercept as \(b = \frac{3}{2}\). So the \(y\)-intercept is the point \((0, \frac{3}{2})\).
06

Finding x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation \(y = \frac{3}{2}\). However, since the equation implies a horizontal line, there is no \(x\)-intercept. The line never crosses the \(x\)-axis.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is crucial in mathematics for understanding lines on a graph. It's denoted by the formula \( y = mx + b \), where:
  • \( m \) represents the slope of the line; it's the rate of change or how steep the line is.
  • \( b \) is the y-intercept; it's the point where the line crosses the y-axis.
This form allows us to quickly assess the direction and position of a line. For a horizontal line, like the one we have here from the equation \( y = \frac{3}{2} \), the slope \( m \) is zero. This means the line doesn't rise or fall, making it parallel to the x-axis. Consequently, the equation simplifies to \( y = 0x + \frac{3}{2} \). Understanding this form makes both manual and graph-assisted equation solving much simpler.
Graphing Equations
Graphing equations is a way to visualize algebraic expressions and their solutions. In the equation \( y = \frac{3}{2} \), graphing involves plotting the horizontal line that lies parallel to the x-axis, cutting through the point where \( y = \frac{3}{2} \).
  • This line is consistent for all values of \( x \) because the slope \( m \) is zero.
  • Think of each \( x \)-coordinate producing the same \( y \)-coordinate, \( y = \frac{3}{2} \).
Drawing lines using the slope-intercept form enables you to identify the nature of the line quickly. In this instance, because the slope is zero, no rise or fall occurs in the line graph. Understanding this will help when distinguishing different types of linear equations on a graph.
Intercepts
Intercepts are essential components of graphing linear equations. They indicate where a line crosses the axes:
  • The y-intercept is found by setting \( x = 0 \) in the equation and solving for \( y \). Here, it's \( \frac{3}{2} \), so the line crosses the y-axis at \((0, \frac{3}{2})\).
  • The x-intercept is found by setting \( y = 0 \) and solving for \( x \).
However, in the equation \( y = \frac{3}{2} \), there's no x-intercept because it's a horizontal line. Such lines are special: they extend indefinitely without crossing the x-axis, highlighting the unique placement of intercepts in linear equations. Comprehending intercepts ensures you can determine how and where a line interacts with the coordinate plane, an invaluable skill in mathematics. Each line behaves differently based on its intercept, adding variety to equation graphing.

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Most popular questions from this chapter

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In the Kingdom of Xyg the tax system works as follows. Someone who earns less than 100 gold coins per month pays no tax. Someone who earns between 100 and 1000 gold coins pays tax equal to \(10 \%\) of the amount over 100 gold coins that he or she earns. Someone who earns over 1000 gold coins must hand over to the King all of the money earned over 1000 in addition to the tax on the first 1000 . (a) Draw a graph of the tax paid \(y\) versus the money earned \(x,\) and give formulas for \(y\) in terms of \(x\) in each of the regions \(0 \leq x \leq 100\), \(100 \leq x \leq 1000,\) and \(x \geq 1000 .\) (b) Suppose that the King of Xyg decides to use the second of these line segments (for \(100 \leq x \leq 1000\) ) for \(x \leq 100\) as well. Explain in practical terms what the King is doing, and what the meaning is of the \(y\) -intercept. \(\Rightarrow\)
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