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Chapter 2: Problem 46

Graph. \(y=x+3\)

Short Answer

Expert verified
Plot points (0, 3) and (1, 4), then draw the line through them.

Step by step solution

01

- Identify the type of equation

The equation given is a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

- Determine the slope and y-intercept

In the equation \( y = x + 3 \), the slope \( m \) is 1 and the y-intercept \( b \) is 3.
03

- Plot the y-intercept

Start by plotting the point at the y-intercept. For \( y = x + 3 \), the y-intercept is 3. Plot the point (0, 3) on the graph.
04

- Use the slope to find another point

The slope \( m = 1 \) means that for every 1 unit increase in \( x \), \( y \) increases by 1 unit. Starting from (0, 3), move 1 unit right to \( x = 1 \) and 1 unit up to \( y = 4 \). Plot the point (1, 4).
05

- Draw the line

With the points (0, 3) and (1, 4) plotted, draw a straight line through these two points. This line represents the graph of the equation \( y = x + 3 \).

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Key Concepts

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

linear equations
Linear equations form one of the most fundamental concepts in algebra. A linear equation is an equation that makes a straight line when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form, which is written as:
y = mx + b.
In this format,
  • y represents the dependent variable
  • x represents the independent variable
  • m represents the slope of the line
  • b represents the y-intercept
Understanding linear equations allows you to describe relationships where one variable depends on another in a proportional and constant manner. These equations are crucial in various fields such as economics, physics, and engineering.
graphing equations
Graphing equations visually represents the relationship described by the equation. For a linear equation like
y = x + 3,
proper graphing involves several steps:
  • First, identify the y-intercept, which is the point where the line crosses the y-axis.
  • Next, use the slope to determine how the line rises or falls as it moves horizontally across the x-axis.
  • With the points plotted, you can draw a straight line through them to represent the equation graphically.
Graphing is a powerful tool for visualizing equations, making it easier to understand the relationship between variables.
slope-intercept form
The slope-intercept form,
y = mx + b,
is a straightforward way to express linear equations. Here's a quick breakdown:
  • The slope (m) indicates the steepness of the line. A higher slope means a steeper line.
  • The y-intercept (b) is the point where the line crosses the y-axis.
For example, in
y = x + 3,
the slope (m) is 1, and the y-intercept (b) is 3. This means the line rises one unit for each unit it moves to the right. By plotting the y-intercept (0, 3) and using the slope to find another point, you can draw the line representing the equation accurately. Understanding the slope-intercept form simplifies the process of graphing linear equations.

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