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Chapter 1: Problem 55

Use the y-intercept and slope to sketch the graph of each equation. $$y-5=0$$

Short Answer

Expert verified
The graph is a horizontal line y = 5.

Step by step solution

01

Identify the equation type

The given equation is in the form of a linear equation: \( y - 5 = 0 \)
02

Simplify the equation

Solve for \( y \): \( y - 5 = 0 \) \( y = 5 \)
03

Interpret the equation

The equation \( y = 5 \) represents a horizontal line where the y-coordinate is always 5.
04

Determine the y-intercept

In the equation \( y = 5 \), the y-intercept occurs where the line crosses the y-axis. This can be seen directly from the equation because \( y = 5 \), thus the y-intercept is at the point (0, 5).
05

Determine the slope

The slope of the line is 0 because it is a horizontal line. Horizontal lines have no vertical change, so the slope \( m \) is \( 0 \).
06

Sketch the graph

Draw the Cartesian plane and plot the y-intercept (0, 5). Then draw a horizontal line through this point because the slope is 0.

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Key Concepts

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

Linear Equations
A linear equation is an equation that forms a straight line when graphed on a coordinate plane. These equations usually take the form \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept of the line.
Linear equations can represent different types of lines, such as sloped, horizontal, or vertical. For example, if we have an equation like \( y = 5 \), it is a linear equation that forms a horizontal line. Understanding the components of the equation helps in sketching its graph accurately.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis on a graph. For the equation \( y = 5 \), the y-intercept is 5. This means that the line will pass through the point (0, 5) on the graph. The y-intercept is important because it provides a starting point for sketching the line.
  • In the equation \( y = mx + b \), the y-intercept is represented by \( b \).
  • For the equation \( y = 5 \), the y-intercept is (0, 5).
Knowing the y-intercept allows us to plot the point on the y-axis and helps in drawing the rest of the line accurately.
Slope
The slope of a line describes how steep the line is. It is usually represented by the symbol \( m \) in the equation \( y = mx + b \). The slope tells us how much the y-coordinate changes for every unit change in the x-coordinate.
  • A horizontal line has a slope of 0.
  • A positive slope means the line goes upwards as it moves from left to right.
  • A negative slope means the line goes downwards as it moves from left to right.
In the equation \( y = 5 \), the line is horizontal, so its slope is 0. This indicates no vertical change as we move along the line, making it a flat horizontal line through the y-intercept.

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Most popular questions from this chapter

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