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Chapter 3: Problem 150

In the following exercises, use slopes and \(y\) -intercepts to determine if the lines are parallel, perpendicular, or neither. $$ y=-1 ; \quad y=2 $$

Short Answer

Expert verified
The lines are parallel.

Step by step solution

01

Identify the Equations

The given equations of lines are: \(y = -1\) and \(y = 2\).
02

Determine the Slopes

These equations are of the form \(y = c\), where \(c\) is a constant. This means both lines are horizontal with a slope of 0.
03

Compare Slopes

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
the slope \textbf{m} is calculated by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Long text would not be good here. so better not elaborate too much here
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It is represented by the 'b' in the equation of a line in the slope-intercept form: \(y = mx + b\). For the equations given \y = -1\ and \y = 2\, the y-intercepts are \ -1\ and \2\ respectively. These points are crucial in understanding the positioning of the lines on a graph. The y-intercept tells us where the line meets the y-axis when \x = 0\.
horizontal lines
Horizontal lines are lines that run straight across from left to right on a graph. In equations of the form \(y = c\), where \ c\ is a constant, the slope is always 0. This means that there is no vertical change as we move along the line. In the given exercise, both \ y = -1 \ and \ y = 2 \ are horizontal lines. They have the same slope of 0 and therefore are parallel to each other.
linear equations
Linear equations are equations that produce a straight line when graphed. They can be written in various forms, such as slope-intercept form \(y = mx + b\), standard form \(Ax + By = C\), and point-slope form \(y - y_1 = m(x - x_1)\). In this exercise, we looked at linear equations in the form \(y = c\). Understanding the different forms helps us to interpret and graph linear relationships effectively.

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