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Chapter 3: Problem 49

Solve equation for \(y\) and then graph it. \(2 x+3 y=-3\)

Short Answer

Expert verified
The solution is \(y = -\frac{2}{3}x - 1\), and the graph is a line crossing the y-axis at -1 with a slope of -2/3.

Step by step solution

01

Rearrange the Equation

The original equation is given by \(2x + 3y = -3\). We want to solve for \(y\), so the first step is to isolate \(3y\) on one side of the equation. Start by subtracting \(2x\) from both sides to get: \(3y = -2x - 3\).
02

Solve for y

To solve for \(y\), divide every term in the equation \(3y = -2x - 3\) by 3. This gives us \(y = \frac{-2x - 3}{3}\). Simplifying the terms, we have \(y = -\frac{2}{3}x - 1\).
03

Identify the y-intercept and Slope for Graphing

The equation \(y = -\frac{2}{3}x - 1\) is in slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(c\) is \(-1\).
04

Plot the y-intercept

On a graph, plot the y-intercept point, which is \((0, -1)\). This is where the line crosses the y-axis.
05

Use the Slope to Find Another Point

From the y-intercept \((0, -1)\), use the slope \(-\frac{2}{3}\) to find another point. The slope means that for every 3 units you move horizontally to the right, you move 2 units down. Starting from \((0, -1)\), moving 3 units right and 2 units down, plot another point at \((3, -3)\).
06

Draw the Line

Connect the points \((0, -1)\) and \((3, -3)\) with a straight line. Extend the line across the graph. Label the line with the equation \(y = -\frac{2}{3}x - 1\).

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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 of a linear equation is one of the most popular ways to express a straight line. This form is given by the equation \(y = mx + c\). Here, \(m\) represents the slope of the line, and \(c\) is the y-intercept. This format is incredibly useful because it allows you to easily identify the slope and y-intercept just by looking at the equation.
It is particularly handy for graphing because you can quickly pick out where the line crosses the y-axis and how steep the line is.
Some key advantages include:
  • Simplified graphing process by identifying the slope and y-intercept.
  • Facilitates easy comparisons between different lines.
  • Enables the quick determination of a line's behavior and direction.
Y-Intercept
The y-intercept of a graph is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + c\), the y-intercept is represented by the term \(c\).
For example, in the equation \(y = -\frac{2}{3}x - 1\), the y-intercept is \(-1\). This means that when \(x = 0\), \(y = -1\).
Knowing the y-intercept is crucial for graphing as it gives you a starting point on the graph. To plot it:
  • Locate \(0\) on the x-axis (vertical line on the graph).
  • Find the corresponding value of \(y\) which is the y-intercept.
  • Place a point where these meet, which is at \((0, -1)\) in our example.
Graphing Lines
Graphing lines using the slope-intercept form is a straightforward process. The equation \(y = mx + c\) not only provides the y-intercept but also gives a clear representation of the slope \(m\), which guides how you plot the line.
To effectively graph a line:
  • First, plot the y-intercept point on the graph.
  • Then, use the slope to determine another point on the line. The slope \(-\frac{2}{3}\) means you move down 2 units for every 3 units you move to the right.
  • From the y-intercept \((0, -1)\), apply the slope to find a second point. In this case, moving 3 units to the right and 2 units down lands on the point \((3, -3)\).
  • Connect these points with a straight line that extends across the graph.
This method ensures accuracy in drawing lines and helps in quickly visualizing the relationship between the variables. Always make sure to label your line with its equation, as this provides clarity for anyone viewing the graph.

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Most popular questions from this chapter

In your own words, what is a function?

iPads. When a student purchased an Apple iPad with \(\mathrm{Wi}_{1}-\mathrm{Fi}+3 \mathrm{G}\) for \(\$ 629.99,\) he also enrolled in a \(250 \mathrm{MB}\) data plan that cost \(\$ 14.95\) per month. a. Write a linear equation that gives the cost for him to purchase and use the iPad for \(m\) months. b. Use your answer to part a to find the cost to purchase and use the iPad for 2 years.

How do we distinguish between a line with positive slope and a line with negative slope?

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((0,0)\) and \((5,-9)\)

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &y=-2 x-9\\\ &2 x-y=9 \end{aligned} $$
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