Problem 23 - Prealgebra

Question

For each of the following equations, complete the given table. $$3 x-2 y=6$$ $$\begin{array}{l|l} \hline x & y \\ \hline-2 & \\ \hline 4 & \\ \hline & 0 \\ \hline & -3 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Fill the table: \((-2, -6), (4, 3), (2, 0), (0, -3)\).

Step 1: Rearrange the Equation

Start by rearranging the given equation to solve for either \( y \). The equation is \( 3x - 2y = 6 \). Solve for \( y \) by first getting \( y \) on one side: \( -2y = 6 - 3x \). Then divide each term by \(-2\): \( y = \frac{3x - 6}{2} \).

Step 2: Fill the Table for Given \( x = -2 \)

Substitute \( x = -2 \) into the rearranged equation to find \( y \): \( y = \frac{3(-2) - 6}{2} = \frac{-6 - 6}{2} = \frac{-12}{2} = -6 \). The value of \( y \) when \( x = -2 \) is \(-6\).

Step 3: Fill the Table for Given \( x = 4 \)

Substitute \( x = 4 \) into the rearranged equation to find \( y \): \( y = \frac{3(4) - 6}{2} = \frac{12 - 6}{2} = \frac{6}{2} = 3 \). The value of \( y \) when \( x = 4 \) is \(3\).

Step 4: Fill the Table for Given \( y = 0 \)

Substitute \( y = 0 \) into the original equation to find \( x \): \( 3x - 2(0) = 6 \Rightarrow 3x = 6 \Rightarrow x = \frac{6}{3} = 2 \). The value of \( x \) when \( y = 0 \) is \(2\).

Step 5: Fill the Table for Given \( y = -3 \)

Substitute \( y = -3 \) into the original equation to find \( x \): \( 3x - 2(-3) = 6 \Rightarrow 3x + 6 = 6 \Rightarrow 3x = 0 \Rightarrow x = \frac{0}{3} = 0 \). The value of \( x \) when \( y = -3 \) is \(0\).

Key Concepts

Solving for Variables

Solving for variables is a fundamental skill in algebra, especially when dealing with linear equations. The goal is to isolate the variable you want to find.
In a linear equation such as \(3x - 2y = 6\), we can solve for one variable in terms of another. This process involves several steps:

• Identify the variable to solve for: You can choose either \(x\) or \(y\). In the given exercise, we solved for \(y\)
• Isolate the variable: Rearrange the equation so that one variable is alone on one side. For example, if solving for \(y\), you would rewrite the equation to \(y = \frac{3x - 6}{2}\)
• Substitute and calculate: Use the rearranged equation to find the value of \(y\) for specific \(x\) values, and vice versa.

This makes it easier to find unknowns given certain conditions, which is crucial for completing tables.

Tables in Mathematics

Tables are an excellent tool for organizing information when working with equations. They allow you to see patterns and relationships between variables clearly.
For the exercise you worked on, a table helped organize \(x\) and \(y\) values efficiently. Here's how you fill such tables:

• Substitute values: Given specific \(x\) or \(y\) values, calculate the corresponding unknown using your formula (like \(y = \frac{3x - 6}{2}\)) and fill them in the table.
• Organize data: Place your known values in one column and the results in the next. This visual representation helps check your work and identify trends or errors.
• Practical applications: Tables can be used beyond math class, such as in data analysis, frequently used in fields like engineering and science.

By using tables, you can avoid confusion and save time when solving problems.

Prealgebra Concepts

Prealgebra covers basic yet essential math concepts that form the foundation for algebra and beyond. Understanding these principles is crucial:

• Equations and balance: The equation \(3x - 2y = 6\) is like a balanced scale. To maintain this balance when solving for a variable, whatever you do on one side, you must do on the other.
• Arithmetic operations: Addition, subtraction, multiplication, and division are the building blocks for solving algebraic equations. They allow you to manipulate equations to isolate variables.
• Substitution: A technique where you replace one variable with its equivalent from another equation or known values. It's an important strategy for evaluating expressions and solving systems of equations.

Grasping these prealgebra concepts ensures you are better prepared for more complex algebraic tasks, like manipulating equations and analyzing functions.