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Chapter 5: Problem 1

What are the slope and the y-intercept of each of these lines? Graph the lines using the slope-intercept method. $$ y=x+1 $$

Short Answer

Expert verified
The slope is 1, and the y-intercept is 1.

Step by step solution

01

Identify the Slope and Y-Intercept

The general form of a linear equation in slope-intercept form is: \[ y = mx + b \]Where \( m \) is the slope and \( b \) is the y-intercept. For the given equation: \[ y = x + 1 \]Compare it with the general form to find the slope \( m \) and y-intercept \( b \). Here, \( m = 1 \) and \( b = 1 \).
02

Plot the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. This happens when \( x = 0 \). Using the identified y-intercept (\( b = 1 \)), plot the point (0, 1) on the graph.
03

Use the Slope to Find Another Point

Slope (\( m \)) describes the change in \( y \) for a unit change in \( x \). With \( m = 1 \), for every increase of 1 unit in \( x \), \( y \) also increases by 1 unit. Starting from the y-intercept (0, 1), move 1 unit right (to \( x=1 \)) and 1 unit up (to \( y=2 \)). Plot this point (1, 2).
04

Draw the Line

Draw a straight line through the two plotted points (0, 1) and (1, 2). This line represents the equation \( y = x + 1 \).

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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 any equation that can be written in the form \( y = mx + b \). This is known as the slope-intercept form. In this form, \( x \) and \( y \) are variables. The equation represents a straight line on a graph. It shows the relationship between \( x \) and \( y \). For example, in the equation \( y = x + 1 \), the graph of this equation will be a straight line.
You can recognize a linear equation if you can see both variables on the graph forming a line. Linear equations are simple and help show the connection between values clearly. This simplicity makes them a fundamental concept in algebra.
Graphing
Graphing is a visual way to represent equations. It's especially useful for understanding linear equations. When graphing the equation \( y = x + 1 \), you plot points on a coordinate plane, which has a horizontal axis (x-axis) and a vertical axis (y-axis).

To graph a linear equation using the slope-intercept method, follow these steps:
  • Identify the slope and y-intercept from the equation.
  • Plot the y-intercept on the graph.
  • Use the slope to find another point from the y-intercept.
  • Draw a straight line through the points.
This method helps in plotting the graph of a linear equation easily and accurately. The visual representation of the graph makes it easier to understand the relationship between variables.
Slope
The slope of a line, denoted as \( m \) in the equation \( y = mx + b \), measures the steepness of the line. It describes how much y changes for a change in x. In the given exercise, the equation \( y = x + 1 \) has a slope of 1. This means for every unit increase in \( x \), \( y \) increases by the same amount.

The slope can be:
  • Positive: If the line goes upwards as it moves from left to right.
  • Negative: If the line goes downwards as it moves from left to right.
  • Zero: If the line is horizontal.
  • Undefined: If the line is vertical.
    • The slope is crucial for understanding how two variables in a linear equation relate to each other.
    Y-Intercept
    The y-intercept is the point where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept form of a linear equation: \( y = mx + b \). In the given equation \( y = x + 1 \), the y-intercept \( b \) is 1. This means the line crosses the y-axis at the point (0, 1).

    To find the y-intercept, set \( x \) to 0 and solve for \( y \). This gives you the point where the line intersects the y-axis. The y-intercept is a simple yet significant concept because it provides a starting point for graphing a linear equation.
    Knowing the y-intercept helps in plotting the line accurately and understanding the behavior of the linear equation at \( x = 0 \).

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    Most popular questions from this chapter

    Change each of the following equations to slope-intercept form, and find the slope. $$ 4 x+2 y=0 $$

    What are the slope and the y-intercept of each of these lines? Graph the lines using the slope-intercept method. $$ y=x-1 $$

    For each of the following, graph the given points. $$ (-4,-2) $$

    b. Find the equation of the line through each pair of points. $$ (1,1) \text { and }(3,0) $$

    For each of the following, graph the given points. $$ (4,0) $$
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