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Chapter 2: Problem 24

Write the equation in the slopeintercept form and then find the slope and \(y\) -intercept of the corresponding line. $$ y-2=0 $$

Short Answer

Expert verified
The given equation is already in the slope-intercept form as \(y = 0x + 2\). The slope of the corresponding line is 0, and the y-intercept is 2.

Step by step solution

01

Rewrite the equation in slope-intercept form

The given equation is: $$ y - 2 = 0 $$ To rewrite it in the slope-intercept form, we simply need to add 2 to both sides of the equation: $$ y = 0x + 2 $$ Now, the equation is in slope-intercept form.
02

Identify the slope of the line

In the slope-intercept form equation, the coefficient of \(x\) represents the slope of the line. In our case, the coefficient of \(x\) is 0. So, the slope (m) of the line is: $$ m = 0 $$
03

Identify the y-intercept of the line

In the slope-intercept form equation, the constant term represents the y-intercept of the line. In our case, the constant term is 2. So, the y-intercept (b) of the line is: $$ b = 2 $$ In conclusion, the slope-intercept form of the given equation is \(y = 0x + 2\). The slope of the corresponding line is 0, and the y-intercept is 2.

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Key Concepts

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

Equation of a line
An equation of a line can be written in different forms, each giving different insights into the properties of the line. One of the most commonly used forms is the slope-intercept form. This form makes it easy to identify the slope and y-intercept of the line. It is generally expressed as \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the y-intercept. In the context of our original problem, re-arranging the equation \( y - 2 = 0 \) into slope-intercept form highlights these line properties for easy interpretation.
Slope
Slope reflects how steep a line is, which is essentially the rate of change or rise over run. When you have the equation of the line in the slope-intercept form, the slope is identified as the coefficient of \( x \). For example, in the equation \( y = 0x + 2 \), the slope \( (m) \) is 0. A slope of 0 signifies a horizontal line, implying that the line is perfectly flat with no tilt either upwards or downwards. This means when you move along the line, the change in \( y \) is zero regardless of the change in \( x \).
Y-intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form of a line \( y = mx + b \), the y-intercept is represented by \( b \). In our exercise, the y-intercept is the value 2 in the equation \( y = 0x + 2 \). This indicates that when \( x = 0 \), \( y = 2 \), positioning the line to intercept through the y-axis at the point (0, 2). This value is crucial as it gives a starting point of the line on a graph.
Linear equations
Linear equations depict relationships between two variables that form a straight line when graphed. They are foundational in algebra and can describe real-world phenomena where one variable changes linearly with another. The standard form of a linear equation comes in many formats, with slope-intercept form \( y = mx + b \) being one of the most straightforward. In our case, \( y - 2 = 0 \) simplifies to \( y = 0x + 2 \) which is a horizontal line. Recognizing these forms aids in solving, interpreting, and graphing linear relationships conveniently in mathematics and its applications.

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