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Chapter 4: Problem 131

In the following exercises, graph each equation. \(2 x+6 y=12\)

Short Answer

Expert verified
Graph from y = -1/3x + 2 over points (0, 2) and (3, 1).

Step by step solution

01

Convert to Slope-Intercept Form

To graph the equation, first convert it to the slope-intercept form, which is: \[ y = mx + b \]Where \( m \) is the slope, and \( b \) is the y-intercept. Start with the given equation:\[ 2x + 6y = 12 \]Solve for \( y \):\[ 6y = -2x + 12 \]\[ y = -\frac{1}{3}x + 2 \]
02

Identify the Slope and Y-Intercept

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

Plot the Y-Intercept

Begin by plotting the y-intercept on the y-axis at the point \( (0, 2) \).
04

Use the Slope to Find Another Point

The slope of \( -\frac{1}{3} \) means that for every 1 unit you move down vertically (negative direction), move 3 units to the right horizontally (positive direction). Starting from \( (0, 2) \), move down 1 unit and 3 units to the right to reach the point \( (3, 1) \).
05

Draw the Line

Plot the second point \( (3, 1) \) on the graph. Draw a straight line through the points \( (0, 2) \) and \( (3, 1) \) to complete the graph of the 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 make graphing linear equations easier, we use the slope-intercept form. The slope-intercept form of a linear equation is given by: \[y = mx + b\] Here, \(m\) represents the slope, and \(b\) represents the y-intercept. The y-intercept is the point where the line crosses the y-axis. Converting an equation to this form can simplify graphing because it helps us easily see both the slope and where the line will cross the y-axis.
Identifying Slope
The slope \(m\) of a line is a measure of how steep the line is. In the equation \(y = -\frac{1}{3}x + 2\), the slope is \(-\frac{1}{3}\). The slope tells us how much the y-coordinate of a point on the line changes for a one-unit change in the x-coordinate. A negative slope means the line slopes downwards from left to right. To identify the slope, look at the coefficient of \(x\) in the equation \(y = mx + b\).
Plotting Y-Intercept
Once you have identified the y-intercept \(b\) from the equation, plotting it is easy. In our example, the y-intercept is 2, which means the line crosses the y-axis at \(y = 2\). To plot it:
  • Start at the origin point (0,0) on a graph.
  • Move vertically up to 2 on the y-axis and plot the point (0,2).
    • This point is crucial because it provides a starting spot for drawing your line.
Using Slope to Find Points
After plotting the y-intercept, use the slope to find other points on the line. For our line, the slope is \(-\frac{1}{3}\), which means:
  • From the y-intercept (0, 2), move down 1 unit (because the slope is negative).
  • Then, move 3 units to the right.
    • This brings you to the point (3, 1). Plot this point. You can continue this process to find more points. Once you have these points, draw a straight line through them. Now you have graphed the equation!

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Most popular questions from this chapter

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x-3 y=8,\) point (4,-1)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(4 x+3 y=6\), point(0,-3)

The city mpg, \(x\), and highway mpg, \(y\), of two cars are given by the points (29,40) and (19,28) . Find a linear equation that models the relationship between city mpg and highway mpg.

Graph the linear inequality \(x+5 y<-5\)

Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)
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