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Chapter 2: Problem 86

Find the slope and y-intercept of each line. Graph the line. $$ -x+3 y=6 $$

Short Answer

Expert verified
Slope: \( \frac{1}{3} \), y-intercept: 2

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. Start by isolating \( y \) in the given equation:\( -x + 3y = 6 \)Add \( x \) to both sides:\( 3y = x + 6 \)Then divide by 3:\( y = \frac{1}{3}x + 2 \)
02

Identify the Slope and y-Intercept

From the equation \( y = \frac{1}{3}x + 2 \), identify the slope (\( m \)) and the y-intercept (\( b \)). Here, the slope \( m \,= \, \frac{1}{3} \) and the y-intercept \( b = 2\).
03

Plot the y-Intercept

To graph the line, start by plotting the y-intercept on the graph. The y-intercept is 2, so place a point at (0, 2) on the y-axis.
04

Use the Slope to Find Another Point

The slope \( \frac{1}{3} \) means that for every 1 unit increase in x, y increases by \( \frac{1}{3} \) units. From the y-intercept (0, 2), move 1 unit to the right (x = 1) and \( \frac{1}{3} \) units up (y = 2.33). Plot this second point at (1, 2.33).
05

Draw the Line

Draw a straight line through the two points (0, 2) and (1, 2.33). This represents the equation of the line.

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Key Concepts

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

Finding Slope
To find the slope of a line, you first need to make sure the equation is in the slope-intercept form. This form looks like this: \( y = mx + b \). Here, \( m \) stands for the slope, and \( b \) is the y-intercept.

In the given equation, \( -x + 3y = 6 \), we start by isolating \( y \).

Let's perform the steps:
  • First, add \( x \) to both sides: \( 3y = x + 6 \).
  • Then, divide everything by 3 to get \( y = \frac{1}{3} x + 2 \).


Now, the equation is in the form of \( y = mx + b \), making it easy to identify the slope, which is \( \frac{1}{3} \). This means the line rises 1 unit for every 3 units it moves horizontally.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. To find the y-intercept, look at the value of \( b \) in the slope-intercept form equation \( y = mx + b \).

In our equation \( y = \frac{1}{3} x + 2 \), the y-intercept \( b = 2 \). This tells us the line crosses the y-axis at the point (0,2).

When graphing, always start by plotting the y-intercept. It's an anchor point for drawing the rest of the line. In this case, you'll place a point at the coordinates (0, 2) on the graph.
Graphing Lines
Graphing a line using the slope-intercept form is straightforward once you have the slope and y-intercept.

Here's a brief guide:
  • First, plot the y-intercept, which is (0, 2), on the graph.
  • Next, use the slope to find another point. The slope \( \frac{1}{3} \) means from the y-intercept, you move 1 unit up for every 3 units you move right.


Starting at (0, 2), move 3 units right to (3, 2). Then move 1 unit up to (3, 3). Mark this point. Now you have two points: (0, 2) and (3, 3).

Draw a straight line through these points. This is the graph of \( y = \frac{1}{3} x + 2 \).

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