Problem 21 - Prealgebra

Question

For each of the following equations, complete the given table. $$y=-4 x$$ $$\begin{array}{c|c} \hline x & y \\ \hline 0 & \\ \hline & 4 \\ \hline & 8 \\ \hline 1 & \end{array}$$

Step-by-Step Solution

Answer

Complete the table as: \( (0, 0), (-1, 4), (-2, 8), (1, -4) \).

Step 1: Calculate y when x = 0

First, substitute 0 for \( x \) in the equation \( y = -4x \). So, \( y = -4(0) = 0 \). This means when \( x = 0 \), \( y = 0 \).

Step 2: Calculate x using y = 4

We need to find \( x \) when \( y = 4 \). Substitute 4 for \( y \) in the equation: \( 4 = -4x \). Solve for \( x \) by dividing both sides by \(-4\): \( x = \frac{4}{-4} = -1 \). So, \( x = -1 \) when \( y = 4 \).

Step 3: Calculate x using y = 8

Substitute 8 for \( y \) in the equation: \( 8 = -4x \). Solve for \( x \) by dividing both sides by \(-4\): \( x = \frac{8}{-4} = -2 \). So, \( x = -2 \) when \( y = 8 \).

Step 4: Calculate y when x = 1

Substitute 1 for \( x \) in the equation: \( y = -4(1) = -4 \). This means when \( x = 1 \), \( y = -4 \).

Key Concepts

The Power of Variable Substitution

Variable substitution is a fundamental method used to solve equations where one variable needs to be isolated or computed when another variable's value is given. It involves replacing one variable with a given number in an equation.

In this exercise, you saw variable substitution come into play when given values of either \( x \) or \( y \) in the equation \( y = -4x \). The steps involved were:

• Identify the known variable and its value, such as \( x = 0 \) or \( y = 4 \).
• Substitute this known value into the equation. For instance, substituting \( x = 0 \) gives \( y = -4(0) = 0 \).
• Solve the equation for the other variable. This gives us a complete understanding of how the values relate to each other in the equation.

By performing variable substitution accurately, you not only find specific values but also understand the relationship between the variables in the equation.

Mastering the Art of Solving for x

When we talk about solving for \( x \), it means finding the value of \( x \) that satisfies an equation when \( y \) is known. This process involves isolating \( x \) on one side of the equation.

To solve for \( x \) in an equation like \( y = -4x \), follow these steps:

• Begin by substituting the known \( y \) value into the equation. For instance, if \( y = 4 \), then use the equation \( 4 = -4x \).
• To isolate \( x \), perform operations that leave \( x \) alone on one side of the equation. In this case, divide both sides by \(-4\), resulting in \( x = \frac{4}{-4} \).
• Simplify the equation. Here, \( x = -1 \) when \( y = 4 \).

This step-by-step process demonstrates how to reverse the operations in the equation to find \( x \) in terms of \( y \). It's crucial for understanding the balance in linear equations.

Solving for y: A Step-by-Step Guide

Solving for \( y \) involves determining the value of \( y \) when \( x \) is known, using the equation \( y = -4x \). This process requires substituting the given \( x \) back into the equation.

Here's how you can solve for \( y \):

• Replace \( x \) with the provided numeric value in the equation. For example, if \( x = 1 \), then the equation becomes \( y = -4(1) \).
• Perform the arithmetic operation as per the equation. In this example, \( y = -4 \times 1 = -4 \).
• Conclude that when \( x = 1 \), then \( y = -4 \). You have solved for another point on the line that this linear equation represents.

As you practice, you'll notice patterns and better understand how changes in \( x \) result in specific changes to \( y \) in the equation, which is central to comprehending linear relationships.

Creating a Table of Values

A table of values is an organized way to list pairs of numbers that the equation satisfies. Such a table not only helps visualize the relationship between variables but also aids in sketching graphs.

To make a table of values for the equation \( y = -4x \), follow these steps:

• Identify key values of \( x \) and calculate corresponding \( y \) values using the equation.
• Record these \( (x, y) \) pairs in two columns: one for \( x \) and one for \( y \).
• For example, if \( x = 0 \), then \( y = 0 \); if \( x = -1 \), then \( y = 4 \); and if \( x = 1 \), then \( y = -4 \).

This table summarizes the solutions, showing how different inputs for \( x \) impact \( y \). It's an essential tool in learning to interpret and graph linear equations and enhances your understanding of their behavior.