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Chapter 3: Problem 34

Graph each equation by using the slope and y-intercept. $$ 3 x+y=-2 $$

Short Answer

Expert verified
Plot y-intercept at \( (0, -2) \) and use slope \(-3\) to plot another point \((1, -5)\). Draw the line through these points.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(y\) on one side of the equation. \[3x + y = -2 \ y = -3x - 2\] Now the equation is in the form \(y = mx + b\).
02

Identify the slope and y-intercept

From the equation \(y = -3x - 2\), identify the slope (\(m\)) and the y-intercept (\(b\)). \[\text{slope} = -3 \ \text{y-intercept} = -2\]
03

Plot the y-intercept

Plot the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For \(y = -3x - 2\), the y-intercept is at \( (0, -2) \). Place a point at \( (0, -2) \).
04

Use the slope to find another point

The slope of -3 means that for every 1 unit the line moves to the right (positive x direction), it moves 3 units down (negative y direction). Starting from the y-intercept \((0, -2)\), move 1 unit to the right to \( (1, -2) \) and then 3 units down to \( (1, -5) \). Plot this point \( (1, -5)\).
05

Draw the line

Draw a straight line through the points \((0, -2)\) and \((1, -5)\). This line represents the graph of the equation \(y = -3x - 2\).

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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 is one of the most popular ways to write a linear equation. It makes it easy to identify the slope (rate of change) and the y-intercept (starting point of the line on the y-axis). The general formula is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
To use this form, you often need to rearrange the given equation to isolate \(y\) on one side.
For instance, given the equation \(3x + y = -2\), you can subtract \(3x\) from both sides to get \(y = -3x - 2\). Now the equation is in slope-intercept form, \(y = mx + b\).
slope
The slope, denoted as \(m\) in the slope-intercept form \(y = mx + b\), measures the steepness and direction of a line.
It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
In the equation \(y = -3x - 2\), the slope is \( -3 \). This means for every step you move to the right (positive x direction), the line goes down by 3 units.
If the slope is positive, the line ascends from left to right. If it is negative, the line descends. A zero slope makes a horizontal line, and an undefined slope makes a vertical line.
y-intercept
The y-intercept, represented as \(b\) in the slope-intercept form \(y = mx + b\), is the point where the line crosses the y-axis.
It tells you where the line starts when \(x = 0\).
In the equation \(y = -3x - 2\), the y-intercept is \( -2 \). This means the line crosses the y-axis at the point \( (0, -2) \).
When graphing, you always begin by plotting the y-intercept on the graph. This gives you a specific point through which the line will pass.
plotting points
Plotting points is a fundamental step in graphing a linear equation.
To begin, you plot the y-intercept. For the equation \(y = -3x - 2\), you start by plotting \( (0, -2) \) on the graph.
Next, use the slope to find another point on the line. From the y-intercept \( (0, -2) \), move one unit to the right and three units down to get the point \( (1, -5) \). Plot this point as well.
Once you have plotted at least two points, draw a straight line through them. This line is the graphical representation of your equation.
By following these steps鈥攑lotting the y-intercept and using the slope to find another point鈥攜ou can easily graph any linear equation.

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Most popular questions from this chapter

The maximum benefit for the heart from exercising occurs if the heart rate is in the target heart rate zone. The lower limit of this target zone can be approximated by the linear equation $$ y=-0.65 x+143 $$ where \(x\) represents age and \(y\) represents heartbeats per minute. (Source: The Gazette.) (a) Complete the table of values for this linear equation. (b) Write the data from the table of values as ordered pairs. (c) Make a scatter diagram of the data. Do the points lie in an approximately linear pattern?

Write the equation of each line with the given slope and \(y\) -intercept. $$ \text { Undefined slope, }(0,5) $$

Evaluate each expression. $$ \frac{5}{8} \cdot \frac{5}{8} $$

Evaluate each expression. $$ 5 \cdot 5 \cdot 5 \cdot 5 $$

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. $$ (8,4), m=1 $$
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