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Chapter 2: Problem 10

Graph \(2 x-3 y=6\)

Short Answer

Expert verified
The graph of the given equation \(2x - 3y = 6\) is a straight line with a slope of \(\frac{2}{3}\) and y-intercept at -2.

Step by step solution

01

Rearrange the equation for y in terms of x

We start by rearranging the given equation \(2x - 3y = 6\) in terms of y. This will allow us to easily identify the slope (m) and y-intercept (b) of the line. Subtract \(2x\) from both sides to get \(-3y = -2x + 6\). Then, divide each term by \(-3\) to solve for y. The rearranged equation is \(y = \frac{2}{3}x - 2\).
02

Identify the slope and y-intercept

In the rearranged equation, the coefficient of x, \(\frac{2}{3}\), is the slope (m) and the constant term, -2, is the y-intercept (b). This tells us that the line rises by 2 units and runs 3 units to the right. It intercepts the y-axis at -2.
03

Plot the y-intercept and slope

Start by plotting the y-intercept (b) at -2 on the y-axis. This is the point where the line crosses the y-axis. Then, from the y-intercept, move up 2 units and 3 units to the right following the slope. Plot a point there. Draw a line through these points to complete the graph of the line.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations can be straightforward once you understand the basics. A linear equation represents a straight line on a graph. The goal is to plot this line based on specific components like the slope and the y-intercept.
To graph a linear equation:
  • First, rearrange the equation to make it easier to identify these components. Usually, this means rewriting it in a form where the equation is as simple as possible.
  • Then, identify the slope and the y-intercept from this rearranged equation. These are your main tools for drawing the line.
  • Once you have these, start by plotting the y-intercept. From there, use the slope to find another point on the line.
  • Draw a straight line through these points, extending it across the graph.
This process transforms the abstract numbers of the equation into a visual representation that is easy to understand and analyze.
Slope-Intercept Form
The slope-intercept form of a linear equation is a standard way of expressing it. It makes plotting the line on a graph much simpler.
This form is given by:
  • \(y = mx + b\)
Where:
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept
Writing an equation as \(y = mx + b\) allows you to quickly see both the slope and the y-intercept.
From the original equation \(2x - 3y = 6\), we rearranged it to get \(y = \frac{2}{3}x - 2\). This direct form provides the tools you need to graph the equation easily.
Y-Intercept
The y-intercept is a critical point when graphing linear equations. It is the point where the line crosses the y-axis of a graph.
In mathematical terms, it's the value of \(y\) when \(x = 0\).
For the equation \(y = \frac{2}{3}x - 2\), the y-intercept is \(-2\).
  • To find it, simply look at the constant term in the slope-intercept form.
  • Plot this point on the y-axis by moving vertically to \(-2\).
This step is crucial because the y-intercept gives you a starting point on the graph. By understanding it, you position the line correctly in its intersection with the y-axis.
Slope
Slope is what gives a line its steepness and direction on a graph. It tells you how the line inclines or declines.
The slope is calculated as the "rise over the run." This means:
  • How many units you move up or down (rise) compared to units moved left or right (run).
From the equation \(y = \frac{2}{3}x - 2\), the slope \(m\) is \(\frac{2}{3}\). This tells us:
  • For every 3 units you move to the right, you move 2 units up.
  • This creates a gentle upward slope as you move along the line.
Understanding slope is essential to accurately plotting the line and gauging its direction and how steep the line is.

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Most popular questions from this chapter

The Stanley Company is coming up with a new cordless travel shaver just before the Christmas holidays. It hopes to sell 10,000 of these shavers in the month of December alone. The manufacturing variable cost is $$\$ 3$$ and the fixed costs $$\$ 100,000$$. If the shavers sell for $$\$ 11$$ each, how many must be produced to break-even?

Find the slope of the line passing through the following pair of points. (4,3) and (-1,3)

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (7,-2) and its \(y\) -intercept is 5 .

Assume a linear relationship holds. In 1975 , an average house in San Jose cost $$\$ 45,000$$ and the same house in 1995 costs $$\$ 195,000$$. Write an equation that will give the price of a house in any year, and use this equation to predict the price of a similar house in the year 2010 .

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (3,5) and (2,-1) .
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