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Chapter 1: Problem 77

Sketch the graph of the equation. Use a graphing utility to verify your result. $$ 2 x-y-3=0 $$

Short Answer

Expert verified
The graph is a line with slope 2 and y-intercept -3. Two points on the line are (0,-3) and (1,-1). A graphing utility would show this line passing through these points.

Step by step solution

01

Identify the form of the equation

The equation is in the standard form for the equation of a line, which is Ax + By + C = 0. Here, A = 2, B = -1, and C = -3. The slope of the line will be -A/B and the y-intercept will be -C/B.
02

Calculate slope and y-intercept

The slope of the line is -A/B = -2/(-1) = 2. The y-intercept is -C/B = 3/(-1) = -3. So, the line intercepts the y-axis at the point (0, -3).
03

Identify another point on the line

To draw the line, we need at least two points. We already have the y-intercept. We can find another point by setting x = 1 in the equation and solving for y. When x = 1, y = 2*1 - 3 = -1. So, another point on the line is (1, -1).
04

Sketch the line

The two points identified, (0,-3) and (1,-1) can be plotted on a graph. Then draw a line that goes through these two points, extending in both directions.
05

Verify the plot using a graphing utility

The equation 2x - y - 3 = 0 can be input into a graphing utility to confirm that the line drawn correctly represents the equation.

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Key Concepts

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

Graphing utility
A graphing utility is a powerful tool that assists in sketching the graph of mathematical equations, like the one we're exploring. These devices or software allow you to input equations and see their graphical representation, which can help verify manual graph sketches.

Using a graphing utility can save time and reduce errors.
  • Graphing calculators and software like Desmos or GeoGebra are examples of these utilities.
  • They often display graphs instantly once the equation is inputted, making it easy to spot mistakes or verify accuracy.
  • Graphing utilities are particularly useful for complex curves or systems of equations where manual graphing would be tedious.
When used alongside manual calculations, they provide a comprehensive understanding of how an equation behaves visually. This dual approach solidifies concepts learned in graphing theory.
Slope-intercept form
The slope-intercept form of a linear equation is one of the most common ways to express a straight line. It's written as y = mx + b, where m is the slope and b is the y-intercept. This form allows for easy graphing by providing direct information about the line's inclination and where it crosses the y-axis.

The slope ( m) represents how steep the line is.
  • A positive slope means the line ascends from left to right, while a negative slope signifies descent.
  • Slope is calculated as the change in y divided by the change in x between two points on the line.
The y-intercept ( b) tells us where the line crosses the y-axis.
  • This point is always of the form (0, b).
Converting the standard form to the slope-intercept form makes it easier to quickly identify these properties and sketch the line on a graph.
Standard form of a line
The standard form of a linear equation is represented as Ax + By + C = 0. This equation format is particularly useful for finding specific properties of the line and for algebraic manipulation. In our exercise, the equation 2x - y - 3 = 0 is in standard form, where A = 2, B = -1, and C = -3.

To convert this to the slope-intercept form, you need to solve for y.
  • Rearrange the equation to isolate y by moving the other terms to one side.
  • This conversion is crucial for interpreting the line graphically, especially when using a graphing utility.
Standard form is advantageous when dealing with multiple equations, especially when solving systems of equations.

Understanding how to recognize and utilize the standard form can enhance your problem-solving toolkit, allowing flexibility in different math contexts.

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Most popular questions from this chapter

Market Equilibrium The supply function for a product relates the number of units \(x\) that producers are willing to supply for a given price per unit \(p .\) The supply and demand functions for a market are $$ \begin{array}{l}{p=\frac{2}{5} x+4} \\ {p=-\frac{16}{25} x+30}\end{array} $$ (a) Use a graphing utility to graph the supply and demand functions in the same viewing window. (b) Use the trace feature of the graphing utility to find the equilibrium point for the market. (c) For what values of does the demand exceed the supply? (d) For what values of does the supply exceed the demand?

Find the limit (if it exists). \(\lim _{\Delta t \rightarrow 0} \frac{(t+\Delta t)^{2}-4(t+\Delta t)+2-\left(t^{2}-4 t+2\right)}{\Delta t}\)

Owning a Business You own two restaurants. From 2001 through 2007 , the sales \(R_{1}\) (in thousands of dollars) for one restaurant can be modeled by $$R_{1}=690-8 t-0.8 t^{2}, \quad t=1,2,3,4,5,6,7$$ where \(t=1\) represents \(2001 .\) During the same seven-year period, the sales \(R_{2}\) (in thousands of dollars) for the second restaurant can be modeled by $$R_{2}=458+0.78 t, \quad t=1,2,3,4,5,6,7$$ Write a function that represents the total sales for the two restaurants. Use a graphing utility to graph the total sales function.

Use a graphing utility to graph the function. Use the graph to determine any x-value(s) at which the function is not continuous. Explain why the function is not continuous at the x-value(s). $$ f(x)=x-2[x] $$

Use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear to be continuous on this interval? Is the function in fact continuous on \([-4,4] ?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{x^{2}+x}{x} $$
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