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Chapter 0: Problem 30

Find the \(x\) - and \(y\) -intercepts if they exist and graph the corresponding line. $$y=\frac{5}{5}$$

Short Answer

Expert verified
The y-intercept is \((0, 1)\), and there is no x-intercept. The line is horizontal at \(y = 1\).

Step by step solution

01

Identify the Function

We are given the equation of the line as: \[y = \frac{5}{5}\] This simplifies to \(y = 1\). The equation \(y = 1\) represents a horizontal line that passes through the point \((x, 1)\) for any value of \(x\).
02

Find the y-intercept

The y-intercept is the point where the line crosses the y-axis. For the equation \(y = 1\), this point is where \(x = 0\). Therefore, directly substitute \(x = 0\) in the equation:\[y = 1\]Thus, the y-intercept is the point \((0, 1)\).
03

Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. This occurs where \(y = 0\). Substitute \(y = 0\) into the equation:\[0 = 1\]This equation is false, indicating that there is no x-intercept. The line is horizontal and never crosses the x-axis.
04

Graph the Line

To graph the line, plot the y-intercept \((0, 1)\) on the coordinate plane. Since the line \(y = 1\) is horizontal, draw a line parallel to the x-axis that passes through this y-intercept point. This line extends infinitely in the positive and negative direction along the x-axis.

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Key Concepts

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

Understanding the X-intercept
The concept of the x-intercept is pivotal when dealing with linear equations. The x-intercept is the point on a graph where the line passes through the x-axis. This happens when the y-value is zero, meaning it's the point
  • Where the graph thoroughly switches from the left to the right side of the x-axis or vice versa.
  • One finds this intercept by setting the y-value to zero in the equation and solving for x.
In the equation given, which simplifies to \[ y = 1 \]:
  • No x-value satisfies this equation when \( y \) is set to zero. This tells us there is no x-intercept.
  • The horizontal line remains steady at \( y = 1 \), never touching the x-axis.
This unique situation is rooted in the nature of horizontal lines.
What is the Y-intercept?
The y-intercept of a line is where the line crosses the y-axis. On the y-axis, all points have an x-coordinate of zero. Recognizing this helps to simplify the task of finding y-intercepts:
  • To identify the y-intercept, set the x-value to zero in your line equation and solve for y.
  • For horizontal lines like \( y = 1 \), the y-intercept is simply \((0, 1)\).
Here, the line crosses the y-axis at this point regardless of the horizontal extension of the line. The y-intercept provides a clear starting point to understand where the line lies on the graph.
Explaining Horizontal Lines
Horizontal lines are fascinating in their simplicity. They have a consistent y-value for any x-value, defined clearly by equations of the form \( y = c \):
  • In our case, the equation \( y = 1 \) tells us that the line stays parallel to the x-axis at \( y = 1 \).
  • These lines never meet the x-axis, explaining the absence of an x-intercept.
One common characteristic is their slope, which is zero. This tells us there's no vertical change as one moves along the line, making horizontal lines straightforward and predictable in their behavior.
Basics of Graphing Lines
Graphing lines effectively starts with identifying key features like intercepts and slopes. In our example of \( y = 1 \), it's crucial to:
  • Plot the y-intercept first. From \((0, 1)\), draw a straight, horizontal line parallel to the x-axis.
  • Understand the line extends endlessly in both the positive and negative x-directions.
This simple visual representation helps to solidify the understanding of how horizontal lines behave on a graph. A thoughtfully plotted graph serves as a vivid geometric illustration of an equation.

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Most popular questions from this chapter

Let \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2}\) be two nonparallel lines \(\left(m_{1} \neq m_{2}\right) .\) What is the \(x\) -coordinate of the point where they intersect?

a. Solve the equation \(x^{2}+2 x=b, b=8\) by first writing in standard form. Now plot both sides of the equation in the same viewing screen \(\left(y_{1}=x^{2}+2 x \text { and } y_{2}=b \right)\) . At what. \(x\) -values do these two graphs intersect? Do those points agree with the solution set you found? b. Repeat (a) for \(b=-3,-1,0,\) and 5

Find the equation of the line that passes through the given point and also satisfies the additional piece of information. Express your answer in slope- intercept form, if possible. (1,4)\(;\) perpendicular to the line \(-\frac{2}{3} x+\frac{3}{2} y=-2\)

Find an equation of a line that passes through the point \((B, A-1)\) and is parallel to the line \(A x+B y=C\). Assume that \(B\) is not equal to zero.

Determine whether each statement is true or false. If the slopes of two lines are -1 and \(1,\) then the lines are perpendicular.
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