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Chapter 4: Problem 309

Graph the line of each equation using its slope and \(y\) -intercept. \(y=-x+3\)

Short Answer

Expert verified
Plot points (0, 3) and (1, 2) and draw a line through them.

Step by step solution

01

- Identify the slope and y-intercept

The equation of the line is in the slope-intercept form, which is given by \(y = mx + b\). In this formula, \(m\) represents the slope, and \(b\) represents the y-intercept. Here, \(m = -1\) and \(b = 3\).
02

- Plot the y-intercept

Locate the y-intercept on the graph. For this equation, the y-intercept \(b = 3\) means the line crosses the y-axis at point (0, 3). Plot this point on the graph.
03

- Use the slope to find another point

The slope \(m = -1\) tells us how to move from the y-intercept to another point on the line. Since the slope is -1, it means you go down 1 unit for every 1 unit you move to the right. Starting from (0, 3), move 1 unit to the right to (1, 3), then 1 unit down to (1, 2). Plot this second point: (1, 2).
04

- Draw the line

Draw a straight line through the two points you have plotted: (0, 3) and (1, 2). This is 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.

Slope-Intercept Form
The slope-intercept form of a linear equation is a way of writing equations so you can easily understand the slope and y-intercept. The general formula is given by: \(y = mx + b\).
Here:
  • \(m\) represents the slope of the line
  • \(b\) represents the y-intercept, the point where the line crosses the y-axis
For example, in the equation \(y = -x + 3\), the slope \(m\) is -1 and the y-intercept \(b\) is 3. This form is helpful because it directly tells you two important things about the line.
Plotting Points
To graph a linear equation, you need at least two points through which you can draw a straight line. Usually, you start with the y-intercept and then use the slope to find another point.

Steps to plot points:
  • Identify the y-intercept from the equation and plot it on the graph. For \(y = -x + 3\), plot (0, 3).
  • Use the slope to find another point starting from the y-intercept. Here, the slope is -1. Move 1 unit to the right and 1 unit down to plot the second point (1, 2).
By plotting the points accurately, you ensure the line represents the equation correctly.
Slope
The slope of a line tells you how steep the line is and the direction it goes. It is often represented by \(m\) in the equation \(y = mx + b\).

The value of the slope can be positive, negative, zero, or undefined:
  • A positive slope means the line goes up from left to right.
  • A negative slope means the line goes down from left to right.
  • A zero slope means the line is horizontal.
  • An undefined slope (division by zero) means the line is vertical.
In the equation \(y = -x + 3\), the slope \(m = -1\) means the line falls 1 unit for every 1 unit it moves to the right.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is represented by \(b\) in the equation \(y = mx + b\).

Steps to identify and plot the y-intercept:
  • Look at the value of \(b\) in the equation. For \(y = -x + 3\), \(b = 3\).
  • Plot this point on the y-axis, which is (0, 3).
The y-intercept is easy to spot because it is where the line intersects the vertical y-axis, making it a key starting point for graphing the line.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(2 x+3 y=6\), containing point (0,5)

Graph the linear inequality \(x \leq-1\).

Graph the linear inequality \(y \leq-3 x\)

Graph the linear inequality \(y>x\)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (3,1) and (2,5)
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