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Chapter 3: Problem 25

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ y=x $$

Short Answer

Expert verified
Slope: 1, Y-Intercept: 0

Step by step solution

01

Identify the Equation Format

The given equation is in the form of \( y = x \). Compare it to the standard slope-intercept form \( y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept. In this equation, there is no additional constant, so it can be rewritten as \( y = 1x + 0 \).
02

Identify the Slope

In the equation \( y = 1x + 0 \), the coefficient of \( x \) is the slope. Therefore, the slope \( m \) is equal to 1.
03

Find the Y-Intercept

The y-intercept \( c \) is the constant term in the equation \( y = mx + c \). In the equation \( y = 1x + 0 \), the constant term is 0, making the y-intercept equal to 0.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a handy way to represent straight lines in algebra. It is given by the equation \( y = mx + c \). This form is particularly useful because it immediately tells us two important characteristics of the line: its slope and its y-intercept.
Understanding this form helps in visualizing how a line behaves in a coordinate plane:
  • \( m \) is the slope of the line. It indicates how steep the line is.
  • \( c \) is the y-intercept. This is the point where the line crosses the y-axis.
By using the slope-intercept form, predicting how changes in the equation affect the line becomes easier, leading to quicker graphing and better comprehension of linear relationships.
Slope
The slope of a line is a measure of how steep the line is. It is usually represented by the letter \( m \) in the slope-intercept form of a line. The slope is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line (commonly known as "rise over run").
The bigger the absolute value of the slope, the steeper the line:
  • If the slope is positive, the line slants upwards from left to right.
  • If the slope is negative, the line slants downwards from left to right.
  • If the slope is zero, the line is horizontal and parallel to the x-axis.
  • An undefined or infinite slope suggests a vertical line.
In our exercise, the slope \( m \) is given as 1, indicating that for each unit increase in \( x \), \( y \) increases by 1 unit. This results in the line slanting upwards at a 45-degree angle.
Y-Intercept
The y-intercept of a line is where it intersects the y-axis. In the equation \( y = mx + c \), the y-intercept corresponds to the constant term \( c \). This point helps in quickly plotting the line since it's where the line engages the y-axis.
The y-intercept is a crucial starting point for graphing because:
  • It's easy to locate on the y-axis as \( c \) directly tells you the intercept without any calculations.
  • This point is often used as a reference for plotting additional points using the slope.
  • It can reveal meaningful contexts in real-world problems where the line models a situation鈥攊n many cases, \( c \) represents an initial value.
In the equation \( y = x \), the y-intercept \( c \) is 0, meaning the line starts exactly at the origin (0,0) on the graph.

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Most popular questions from this chapter

A sporting goods manufacturer allocates at least \(2,400\) units of production time per day to make baseballs and footballs. It takes 20 units of time to make a baseball and 30 units of time to make a football. If \(x\) represents the number of baseballs made and \(y\) represents the number of footballs made, the graph of \(20 x+30 y \geq 2,400\) shows the possible ways to schedule the production time. Graph the inequality. Then find three possible combinations of production time for the company to make baseballs and footballs.

Two planes are \(6,000\) miles apart, and their speeds differ by \(200 \mathrm{mph}\). They travel toward each other and meet in 5 hours. Find the speed of the slower plane.

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Find the slope of each line. See Examples 4 and \(5 .\) $$ y=8 $$

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