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Chapter 4: Problem 357

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(y=5 ; \quad y=1\)

Short Answer

Expert verified
The lines are parallel because they both have a slope of \(0\).

Step by step solution

01

Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + c \] where \(m\) is the slope and \(c\) is the y-intercept.
02

Analyze the Given Equations

Rewrite the given equations to compare their slopes and y-intercepts. The first equation is: \[ y = 5 \] The second equation is: \[ y = 1 \]
03

Determine the Slopes

Both equations are horizontal lines. A horizontal line can be represented in the form: \[ y = c \] where \(c\) is a constant. The slope \(m\) of a horizontal line is always \(0\).
04

Compare the Slopes

Since the slope \(m\) for both lines is zero: \[ m_1 = 0 \] \[ m_2 = 0 \] The slopes are equal.
05

Conclusion on Line Parallelism

Two lines are parallel if and only if their slopes are equal. Since both lines have the same slope of \(0\), they are parallel.

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Key Concepts

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

'Slope-Intercept Form'
When we talk about the equation of a line, the slope-intercept form is probably the most common way to express it. This form is given by the equation:
\( y = mx + c \)
Here, \(y\) represents the dependent variable, \(x\) represents the independent variable, \(m\) is the slope of the line, and \(c\) is the y-intercept, the point where the line crosses the y-axis.
Let's break down these parts:
  • Slope (\(m\)): This tells us how steep the line is. A higher absolute value of \(m\) means a steeper slope.
  • Y-Intercept (\(c\)): This is where the line crosses the y-axis. It's the value of \(y\) when \(x\) is 0.

For instance, in the equation \( y = 2x + 3 \), the slope is 2, and the y-intercept is 3.
'Horizontal Lines'
A horizontal line is special because it runs straight across from left to right. It's perfectly flat. An equation of a horizontal line is written as:
\( y = c \)
Here, \(c\) is a constant. This means that no matter what \(x\) is, \(y\) always remains the same.
Let's consider the given equations:
  • First equation: \( y = 5 \)
  • Second equation: \( y = 1 \)

Both of these are horizontal lines. If you plot these on a graph, the lines will be parallel to the x-axis.
Horizontal lines don't have a slope in the traditional sense. The slope \(m\) of a horizontal line is always 0. This is because there is no change in \(y\) as \(x\) changes, leading to zero rise over run.
'Slope Comparison'
To determine if two lines are parallel, we need to compare their slopes.
Two lines are parallel if and only if their slopes are equal.
Let's revisit the given equations:
  • First equation: \( y = 5 \)
  • Second equation: \( y = 1 \)

We've already determined that both of these lines are horizontal, and the slope of a horizontal line is 0.
So, the slope \(m_1\) of the first line is 0, and the slope \(m_2\) of the second line is also 0.
Since \(m_1 = m_2\), the slopes are equal.
Therefore, we can conclude that the lines are parallel.
Remember, the key takeaway here is:
  • Equal slopes indicate parallel lines.
  • For horizontal lines, the slope is always 0.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. \(m=\frac{5}{6},\) containing point (6,7)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(y+2=0\), point (3,-3)

Graph the linear inequality \(y>x\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+5 y=6,\) point(0,0)

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (2,7) and (3,8)
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