Problem 20 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The completed table is: \( x=3, y=1 \); \( x=-2, y=-4 \); \( x=4, y=2 \); \( x=8, y=6 \).

Step 1: Understand the Given Equation

We are given the equation \(x-y=2\). This is a linear equation in two variables, \(x\) and \(y\), which implies a straight line when plotted on a graph.

Step 2: Substitute and Solve for y (First Case)

We need to find \(y\) for \(x=3\). Substitute \(x=3\) into the equation \(x-y=2\), resulting in \(3-y=2\). To solve for \(y\), isolate \(y\): \(y=3-2=1\).

Step 3: Substitute and Solve for y (Second Case)

Now find \(y\) for \(x=-2\). Substitute \(x=-2\) into the equation \(x-y=2\), so \(-2-y=2\). Solving for \(y\), we get \(-y=4\), hence \(y=-4\).

Step 4: Substitute and Solve for x (Third Case)

We need to find \(x\) when \(y=2\). Substitute \(y=2\) into the equation \(x-y=2\), giving us \(x-2=2\). Solving for \(x\), \(x=2+2=4\).

Step 5: Substitute and Solve for x (Fourth Case)

Lastly, find \(x\) when \(y=6\). Substitute \(y=6\) into the equation, resulting in \(x-6=2\). Solving for \(x\), \(x=8\).

Step 6: Fill in the Table

Using the computed values, fill in the table: \[\begin{array}{l|l}\hline x & y \\hline 3 & 1 \\hline -2 & -4 \\hline 4 & 2 \\hline 8 & 6 \\hline\end{array}\]

Key Concepts

Understanding Two Variables

When dealing with linear equations, understanding that they often contain two variables is essential. In this context, the equation is \( x - y = 2 \). Our equation involves two variables: \( x \) and \( y \). Each represents a different unknown that we need to determine based on the conditions provided in a problem.
To decipher the mystery of these variables, one needs to recognize that they interact with each other. Their values depend on each other to satisfy the equation. This relationship is the foundation for figuring out their values, as one variable can often be expressed in terms of the other.
Linear equations with two variables typically graph as straight lines in a two-dimensional space. The values of \( x \) and \( y \) serve as coordinates that help confirm their positions on this line. This interplay highlights why mastering two-variable equations can be a valuable skill in both mathematics and various real-world applications.

The Process of Solving Equations

When tackling equations like \( x - y = 2 \), the process of finding both \( x \) and \( y \) is crucial. This task involves performing mathematical operations to isolate one variable at a time.
The main goal is to solve for one variable when given a value for the other. This process generally involves:

• Substituting the known value of one variable into the equation
• Rearranging and simplifying the equation
• Solving for the remaining variable

For instance, if you know the value of \( x \), you can substitute this into the equation and solve for \( y \). Similarly, if \( y \) is known, you can determine \( x \). This step-by-step approach ensures that every aspect of the equation is examined, making the solution process both systematic and logical.

Exploring the Substitution Method

The substitution method is a common technique used to solve systems of equations, like the given equation \( x - y = 2 \). This method is particularly useful in equations with two variables.Here's how it works:

• Choose an equation from the system, and solve it for one variable in terms of the other.
• Substitute this expression into the other equation.
• This substitution yields an equation with a single variable, facilitating an easier solution.
• Finally, substitute back to find the second variable.

The substitution method is incredibly effective because it transforms a potentially complex problem into a simpler, more manageable one. With practice, this method becomes a powerful tool in solving a wide range of mathematical problems, offering clarity in how variables relate to each other within an equation.

Utilizing a Table of Values

A table of values is a helpful visual tool to organize and understand the solutions for equations like \( x - y = 2 \). This table allows for a quick overview of how different values of one variable affect the other variable in the equation.Using a table of values involves:

• Entering known values for one variable in the table.
• Calculating the corresponding values for the second variable using the equation.
• Filling in the table to represent these variable pairs clearly.

Such a table not only aids in cross-verifying the solutions but also offers a clear graphical illustration of the relationship between variables. It assists in verifying that all calculated values satisfy the given equation. Moreover, the table serves as a guide for predicting values without solving the equation repeatedly. Utilizing this organizational tool can greatly enhance one's ability to analyze equations systematically.