Problem 7 - Precalculus

Question

(a) Solve the equation \(2 x-3 y=-3\) for \(y\) and then complete the following table. $$\begin{array}{lccccc} \hline x & -6 & -3 & 0 & 3 & 6 \\ \hline y & & & & & \\ \hline \end{array}$$ (b) Use your table from part (a) to graph the equation \(2 x-3 y=-3\)

Step-by-Step Solution

Answer

Solve \(y = \frac{2x + 3}{3}\), complete the table, and plot the graph.

Step 1: Solve for y

We start by rearranging the equation \(2x - 3y = -3\) to solve for \(y\). First, isolate the term containing \(y\) by adding \(3y\) to both sides: \(2x = 3y - 3\). Then, add 3 to both sides to get: \(2x + 3 = 3y\). Finally, divide everything by 3 to solve for \(y\): \(y = \frac{2x + 3}{3}\).

Step 2: Create a Table of Values

Using the equation from Step 1, substitute each value of \(x\) from the table into \(y = \frac{2x + 3}{3}\) to find the corresponding value of \(y\). - For \(x = -6\), \(y = \frac{2(-6) + 3}{3} = \frac{-12 + 3}{3} = -3\).- For \(x = -3\), \(y = \frac{2(-3) + 3}{3} = \frac{-6 + 3}{3} = -1\).- For \(x = 0\), \(y = \frac{2(0) + 3}{3} = \frac{3}{3} = 1\).- For \(x = 3\), \(y = \frac{2(3) + 3}{3} = \frac{6 + 3}{3} = 3\).- For \(x = 6\), \(y = \frac{2(6) + 3}{3} = \frac{12 + 3}{3} = 5\).These values complete the table.

Step 3: Complete the Table

Fill in the values of \(y\) in the table as follows:\[\begin{array}{lccccc} \hline x & -6 & -3 & 0 & 3 & 6 \ \hline y & -3 & -1 & 1 & 3 & 5 \ \hline \end{array}\]

Step 4: Plot the Graph

Use the completed table of values to plot the points \((-6, -3)\), \((-3, -1)\), \((0, 1)\), \((3, 3)\), and \((6, 5)\) on a graph. Connect these points with a straight line to represent the equation \(2x - 3y = -3\). Ensure the line extends in both directions beyond the plotted points.

Key Concepts

Solving for y

To solve for \( y \) in a linear equation like \( 2x - 3y = -3 \), we need to rearrange the equation so that \( y \) is isolated on one side. This means we'll manipulate the equation step by step until \( y \) stands alone on one side of the equals sign. Here’s how you can do it: 1. Start with the original equation: \( 2x - 3y = -3 \). 2. Move the term \( 2x \) to the opposite side by subtracting \( 2x \) from both sides: \( -3y = -2x - 3 \). 3. Divide every term by \(-3\) to solve for \( y \): \[y = \frac{-2x - 3}{-3} = \frac{2x + 3}{3}.\]This equation, \( y = \frac{2x + 3}{3} \), is now in the form that clearly shows how \( y \) changes based on different \( x \) values.

Graphing a Line

Once we have the equation in the form \( y = \frac{2x + 3}{3} \), we can graph this line by plotting several points. These points are derived from substituting different values for \( x \) into the equation.Imagine setting up a small "coordinate party." Here’s what we do:- Choose a range of \( x \) values. These are like your party guests.- Use each \( x \) value to calculate a corresponding \( y \) value using the formula we found for \( y \).- Each \( (x, y) \) pair gives you a point you can plot on a graph.After plotting several points (like \((-6, -3)\), \((-3, -1)\), and so on), connect these points with a straight line. Make sure to extend the line across the graph. This line represents every possible \( (x, y) \) pair that satisfies the equation \( 2x - 3y = -3 \). A straight line tells us that our relationship between \( x \) and \( y \) is linear—consistent and unchanging.

Table of Values

A table of values is a simple, organized way to show specific \( (x, y) \) pairs that satisfy a linear equation. By using our formula \( y = \frac{2x + 3}{3} \), we substitute various \( x \) values to find matching \( y \) outputs.Here’s how the table works:- Choose a set of \( x \) values, such as \(-6, -3, 0, 3, 6\). Think of these as instructions or inputs.- For each \( x \), calculate \( y \) using the equation.For example:

• When \( x = -6 \), \( y = -3 \).
• When \( x = -3 \), \( y = -1 \).
• When \( x = 0 \), \( y = 1 \).
• When \( x = 3 \), \( y = 3 \).
• When \( x = 6 \), \( y = 5 \).

These \( (x, y) \) pairs fill the table. It's like a list that both clarifies the relationships between \( x \) and \( y \), as well as serves as a guide for graphing. A completed table not only helps ensure your math is correct but also visually supports the plotting of points on a graph.