91Ó°ÊÓ

In algebra, variables are symbols that stand in for unknown values. They allow us to generalize mathematical expressions and solve problems more flexibly. In the given exercise, we have three variables: \(x\), \(y\), and \(z\). Each represents a specific quantity that can change, and these quantities are related to each other through algebraic expressions.
For instance, \(x\) is defined as twice the quantity of \(y\), which can be written mathematically as \(x = 2y\). Similarly, \(y\) is three times the value of \(z\), expressed as \(y = 3z\).
The purpose of using variables is to simplify complex problems and describe the relationships between different parts of the problem. By substituting these relationships using variables, we can solve for one variable in terms of another, showing the interconnectedness of the problem's components.

Equation solving is at the heart of this exercise. It involves finding the value of unknown variables that make the equation true. When we're given conditions that describe how variables relate to each other, our job is to express one variable in terms of others by manipulating these equations.
In this scenario, we began by understanding the relationships: \(y = 3z\) and \(x = 2y\). The goal is to express \(x\) in terms of \(z\). By using these equations, we systematically replace and rearrange terms to unveil the desired relationship.

• First, write each relationship using equations.
• Next, substitute where necessary to eliminate one variable.
• Finally, simplify to express one variable solely in terms of another.

Equation solving requires logical thinking and accuracy to ensure each step follows mathematically sound operations.

The substitution method is a common technique in solving simultaneous equations. It involves replacing a variable with an equivalent expression, which simplifies solving the equation.
In this exercise, we used substitution by taking the expression for \(y\) from one equation and placing it into another equation that involves \(y\).

• Start with identifying which variable to substitute. Here \(y = 3z\).
• Use this relationship in the equation \(x = 2y\).
• Replace \(y\) in \(x = 2y\) with \(3z\), leading to \(x = 2(3z)\).

This method helps efficiently reduce equations to a simpler form, making it easier to relate variables and find the solution. After substitution, simplify the expression to reveal \(x = 6z\), explaining how \(x\) and \(z\) are directly related.