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Chapter 2: Problem 52

Graph. \(y+3 x=1\)

Short Answer

Expert verified
Rewritten as \(y = -3x + 1\), plot (0, 1) and (1, -2), and draw the line.

Step by step solution

01

- Rewrite the equation in slope-intercept form

First, rearrange the given equation into the slope-intercept form, which is \(y = mx + b\). Start with the given equation: \(y + 3x = 1\). Subtract \(3x\) from both sides to get \(y = -3x + 1\).
02

- Identify the slope and y-intercept

The slope-intercept form is now \(y = -3x + 1\). Identify the slope (m) and the y-intercept (b). Here, \(m = -3\) and \(b = 1\).
03

- Plot the y-intercept

To graph the line, start by plotting the y-intercept \(b = 1\) on the y-axis. This point is (0, 1).
04

- Use the slope to find another point

Using the slope \(m = -3\), which means a rise of -3 and a run of 1. From the y-intercept (0, 1), move down 3 units and right 1 unit to find another point on the line at (1, -2).
05

- Draw the line

Draw a line through the two points (0, 1) and (1, -2). This line is the graph of \(y = -3x + 1\) which satisfies the given equation.

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Key Concepts

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

slope-intercept form
To graph a linear equation, it's helpful to rewrite it in the slope-intercept form. The slope-intercept form is represented by the equation \(y = mx + b\). Here, the letter 'm' stands for the slope, and 'b' is the y-intercept.
This form makes graphing easy as it directly tells us the slope of the line and where it intersects the y-axis.
slope
The slope of a line describes how steep the line is. In the equation \(y = mx + b\), the 'm' represents this slope.
The slope is calculated as the 'rise' over the 'run', meaning the change in the y-values divided by the change in the x-values between two points on the line.
For the equation \(y = -3x + 1\), the slope is -3. This tells us that the line goes down 3 units for every 1 unit it goes to the right.
y-intercept
The y-intercept is where the line intersects the y-axis. It's the 'b' in the equation \(y = mx + b\).
The y-intercept is the value of y when x is 0. In our example equation, \(y = -3x + 1\), the y-intercept is 1. This point (0, 1) tells us exactly where to start plotting the line on the graph.
plotting points
Plotting points is essential to graphing any linear equation. Start with the y-intercept point. Then use the slope to find additional points.
From our y-intercept (0, 1), we use the slope -3 to find another point. Since our slope is -3, we move down 3 units and 1 unit to the right, which lands us at the point (1, -2).
Connecting these points with a straight line gives us the graph of the equation \(y = -3x + 1\).

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