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Chapter 2: Problem 148

What can you say about two lines that have equal slopes and equal \(y\) -intercepts?

Short Answer

Expert verified
The lines are the same.

Step by step solution

01

Identify the Equation of a Line

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

Compare Slopes

Two lines with the same slope are parallel. If the slopes \( m_1 \) and \( m_2 \) of the two lines are equal, then: \[ m_1 = m_2 \]
03

Compare Y-Intercepts

If the two lines also have the same \( y \)-intercept \( b \), then: \[ b_1 = b_2 \]
04

Determine Line Properties

When two lines have both the same slope and the same \( y \)-intercept, their equations are identical. This means the two lines are not just parallel; they are the same line.

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Key Concepts

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

Equation of a Line
To start understanding lines in geometry, we need to grasp the concept of the equation of a line. The slope-intercept form is one of the most common ways to write a line equation. It is written as: \[ y = mx + b \] Where:
  • \( y \) is the dependent variable or the output value.
  • \( x \) is the independent variable or the input value.
  • \( m \) represents the slope, which indicates how steep the line is.
  • \( b \) is the y-intercept, showing where the line crosses the y-axis.
Understanding each part of this equation is key, as it allows us to define how a line behaves and looks on a graph.
Parallel Lines
Parallel lines are an interesting concept in geometry. These lines never intersect, no matter how far they extend on the plane. For lines to be parallel, they must have the same slope. This means in their slope-intercept form: \[ y = m_1x + b_1 \] and \[ y = m_2x + b_2 \] We have that \( m_1 = m_2 \). This is the condition for parallelism. The different y-intercepts \( b_1 \) and \( b_2 \) ensure the lines do not overlap.
Identical Lines
Identical lines are essentially the same line on a graph. They overlap completely and are indistinguishable from one another. For two lines to be identical, they need to have not just the same slope but also the same y-intercept. In equation form, if \[ y = m_1x + b_1 \] and \[ y = m_2x + b_2 \], then \( m_1 = m_2 \) and \( b_1 = b_2 \). This means that every corresponding point on one line is also a point on the other line. Identical lines will coincide at all points, effectively making them the same line.

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Most popular questions from this chapter

Driving to School The time \(t\) that it takes to get to school varies inversely with your average speed \(s\). (a) Suppose that it takes you 40 minutes to get to school when your average speed is 30 miles per hour. Express the driving time to school in terms of average speed. (b) Suppose that your average speed to school is 40 miles per hour. How long will it take you to get to school?

Find the slope and y-intercept of each line. Graph the line. $$ y=\frac{1}{2} x+2 $$

Find the slope and y-intercept of each line. Graph the line. $$ x+y=0 $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The lengths of the legs of a right triangle are \(a=8\) and \(b=15\). Find the hypotenuse.

Find the slope and y-intercept of each line. Graph the line. $$ 3 x+2 y=6 $$
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