Problem 22 - Prealgebra

Question

For each of the following equations, complete the given table. $$y=5 x$$ $$\begin{array}{l|l} \hline x & y \\ \hline & 5 \\ \hline & 0 \\ \hline-1 & \\ \hline 2 & \end{array}$$

Step-by-Step Solution

Answer

The completed table is: \((1, 5), (0, 0), (-1, -5), (2, 10)\).

Step 1: Understand the Equation

The given equation is \(y = 5x\). This means to find the value of \(y\), we multiply \(x\) by 5.

Step 2: Fill in the Table for First Row

We need to find \(x\) when \(y = 5\). Using the equation \(y = 5x\), set \(y\) equal to 5: \(5x = 5\). Solve for \(x\): \(x = \frac{5}{5} = 1\). Thus, when \(y = 5, x = 1\).

Step 3: Fill in the Table for Second Row

We need to find \(x\) when \(y = 0\). Set \(y\) equal to 0: \(5x = 0\). Solve for \(x\): \(x = \frac{0}{5} = 0\). Thus, when \(y = 0, x = 0\).

Step 4: Fill in the Table for Third Row

Here we need to find \(y\) when \(x = -1\). Substitute \(x = -1\) into the equation \(y = 5x\): \(y = 5(-1) = -5\). So, when \(x = -1, y = -5\).

Step 5: Fill in the Table for Fourth Row

Lastly, find \(y\) when \(x = 2\). Substitute \(x = 2\) into the equation \(y = 5x\): \(y = 5(2) = 10\). Therefore, when \(x = 2, y = 10\).

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation that represents the repeated addition of a number. In the context of linear equations, like the equation \(y = 5x\), multiplication helps in scaling the value of \(x\) to find \(y\). For instance, if \(x = 2\), multiplying it by 5 results in \(y = 10\).
This shows that multiplication doesn't only enlarge numbers; it also helps in understanding relationships between variables.

• It represents repeated addition: \(5 \times 2 \equiv 2 + 2 + 2 + 2 + 2\).
• It scales numbers, meaning it can increase or decrease them.
• It is a tool to find solutions in equations quickly by applying direct calculations.

Solving Equations

Solving linear equations involves finding the values of the variables that satisfy the equation. In our exercise, we solve for \(x\) or \(y\) using the equation \(y = 5x\). This can either mean finding a direct value of \(y\) from a given \(x\) or vice versa.
Here's how it works:

• Identify the unknown you need to solve for, whether it's \(x\) or \(y\).
• If given \(y\), set up the equation and solve for \(x\): \(y = 5x\) implies rearranging to \(x = \frac{y}{5}\).
• If given \(x\), substitute it into the equation to find \(y\): simply \(y = 5x\).

Solving equations is a crucial skill, as it builds the foundation for problem-solving in algebra and beyond. Understanding how to manipulate an equation gives you the power to tackle a wide range of math problems.

Mathematical Tables

Mathematical tables are handy tools for organizing data and showing relationships between different quantities. In the provided exercise, the table illustrates the link between \(x\) and \(y\) as derived from the equation \(y = 5x\).
Tables serve several purposes:

• They allow for easy comparison of values and quick analysis of output for different inputs.
• In learning environments, tables help students to systematically fill in values and reinforce understanding of the relationships being studied.
• They visually display results, making it easier to identify patterns and trends.

By completing a table, like in our exercise, students practice calculating various outcomes (\(y\) values) based on different inputs (\(x\) values), thereby cementing their understanding of how equations work.