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Chapter 3: Problem 41

Construct a table of solutions and then graph equation. \(y=2 x-3\)

Short Answer

Expert verified
Create a table of solutions, plot points on a graph, and connect them with a straight line.

Step by step solution

01

Choose Values for x

To construct a table of solutions for the equation, start by picking several values for the variable \(x\). Common choices are integer values such as \(-2, -1, 0, 1, 2\).
02

Calculate Corresponding y Values

For each chosen \(x\) value, substitute it into the equation \(y = 2x - 3\) to find the corresponding \(y\) value. For example, if \(x = -2\), then \(y = 2(-2) - 3 = -4 - 3 = -7\). Repeat for other \(x\) values.
03

Create a Table of Solutions

List each \(x\) value along with its corresponding \(y\) value in a table format. For example:\[\begin{array}{|c|c|}\hlinex & y = 2x-3 \\hline-2 & -7 \-1 & -5 \0 & -3 \1 & -1 \2 & 1 \\hline\end{array}\]
04

Plot the Points on a Graph

Using the table of solutions, plot the points \((-2, -7), (-1, -5), (0, -3), (1, -1), (2, 1)\) on a coordinate plane. Each point corresponds to an \((x, y)\) pair from the table.
05

Draw the Line

Connect the plotted points with a straight line. This line represents the graph of the equation \(y = 2x - 3\). Ensure the line extends in both directions beyond the plotted points to accurately represent all solutions to the equation.

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Key Concepts

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

Graphing
Graphing an equation involves visually representing a mathematical relationship between variables on a coordinate plane. When you graph the equation \(y = 2x - 3\), you are looking at a straight line, since this is a linear equation. To graph effectively, follow these steps:
  • Start by preparing a table of solutions, which lists pairs of \(x\) and \(y\) values that satisfy the equation.
  • Next, plot these points on the graph. Your graph is a visual representation of how \(y\) changes with \(x\), and plotting each pair will help situate this relationship onto your paper or screen.
  • Finally, connect the points with a straight line. In the case of \(y = 2x - 3\), this line will have a slope of 2 and will cross the y-axis at -3, indicating where the line intercepts the \(y\)-axis.
Graphing gives you a visual picture, helping you understand how the values of \(x\) affect \(y\) in the equation.
Cartesian Plane
The Cartesian plane, or coordinate plane, is a two-dimensional space formed by two perpendicular axes, which are denoted as the \(x\)-axis (horizontal) and \(y\)-axis (vertical). The Cartesian plane is fundamental in graphing because:
  • It provides a way to plot points represented as ordered pairs \((x, y)\).
  • The intersection of the \(x\)-axis and \(y\)-axis is known as the origin, where both \(x\) and \(y\) are zero \((0, 0)\).
  • Each point on the plane defines a unique location determined by its coordinates.
  • The plane is divided into four quadrants, which help signify the sign (+/-) of its coordinates depending on whether \(x\) and \(y\) are positive or negative.
The Cartesian plane serves as the backdrop for drawing the graph of linear equations like \(y = 2x - 3\). As you plot points and draw lines, it helps illustrate the relationship laid out by the equation.
Table of Solutions
A table of solutions is a simple yet important tool in the process of graphing an equation. Here's how you utilize it:
  • Begin by selecting several easy values for \(x\). Often, integer values like -2, -1, 0, 1, 2 are chosen for simplicity.
  • Substitute these \(x\) values into the linear equation to compute corresponding \(y\) values. For \(y = 2x - 3\), each substitution yields a different point on the graph.
  • Once calculated, organize these \(x\) and \(y\) pairs into the table. This setup provides a clear view of points that can be directly plotted on a graph.
A table of solutions is not only a reflective list of points; it serves as a step-by-step guide that simplifies understanding, especially as you move from choosing \(x\)-values to calculating \(y\)-values, and eventually plotting them to form a graph. It's a vital step to ensure all relationships in the equation are clearly represented.

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Most popular questions from this chapter

A dentist's office schedules 1 -hour long appointments for adults and \(\frac{3}{4}\) -hour long appointments for children. The appointment times do not overlap. Let \(c\) represent the number of appointments scheduled for children and a represent the number of appointments scheduled for adults. The graph of \(\frac{3}{4} c+a \leq 9\) shows the possible ways the time for seeing patients can be scheduled so that it does not exceed 9 hours per day. Graph the inequality. Label the horizontal axis \(c\) and the vertical axis \(a .\) Then find three possible combinations of children/adult appointments.

Find the slope of each line. See Examples 4 and \(5 .\) $$ x=4 $$

Find the slope of each line. See Examples 4 and \(5 .\) $$ y-9=0 $$

Let \(f(x)=x-2\) and \(g(x)=3 x .\) Find \(f(g(6))\).

Production Costs. A television production company charges a basic fee of \(\$ 5,000\) and then \(\$ 2,000\) an hour when filming a commercial. a. Write a linear equation that describes the relationship between the total production costs \(c\) and the hours \(h\) of filming. b. Use your answer to part a to find the production costs if a commercial required 8 hours of filming
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