Problem 33 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ x+y=z, \text { for } y $$

Step-by-Step Solution

Answer

Answer: The expression for y in terms of x and z is y = z - x.

Step 1: Write down the given equation

The given equation is: $$ x+y=z $$

Step 2: Isolate y

To isolate y, we'll subtract x from both sides of the equation to remove x from the left side: $$ y=z-x $$

Step 3: Final expression for y

The isolated expression for y in terms of x and z is: $$ y=z-x $$

Key Concepts

Solving Equations

When solving equations, our goal is to find the value or expression for an unknown variable. Equations like the one in the original exercise are statements of equality between two expressions. There are different types of equations, such as linear, quadratic, and others, but the process of solving them often follows some basic steps.

To solve an equation, we often perform operations like addition, subtraction, multiplication, or division to both sides. This helps us maintain the balance of the equation while simplifying it to reveal the unknown variable's value. In simple terms, think of an equation like a balanced scale, where whatever you do to one side, you must also do to the other to keep it balanced.

• Identify the equation you need to solve.
• Determine what the unknown variable is.
• Use arithmetic operations to isolate the variable.

Solving equations can start as a simple task but may become more complex with additional terms and rules. Always remember, however, the logic behind solving remains the same: balancing both sides to reveal the unknown.

Variables

Variables are symbols that represent unknown quantities in an equation. In algebra, these are typically alphabetic characters like \( x \), \( y \), or \( z \). In our exercise, \( y \) is the variable we aim to solve for.

Variables are crucial because they allow us to model real-world situations and solve problems without knowing all of the information upfront. They are placeholders that can change depending on the context of the problem. Think of variables as containers that can hold different values, allowing us flexibility and the ability to manipulate equations to find these values.

• Variables can represent numbers, parameters, or expressions.
• They allow the abstraction of problems, offering generalized solutions.

Understanding variables is an essential part of learning algebra, as they pave the way for solving complex equations and inequalities.

Isolation Technique

The isolation technique is a fundamental algebraic method used to solve equations with one or more unknown variables. This method involves rearranging the equation to get the variable of interest alone on one side. In our exercise, we aim to isolate \( y \) by getting it by itself on one side of the equation.

To isolate \( y \) in the equation \( x + y = z \), we subtract \( x \) from both sides, thus removing \( x \) from the left side of the equation. This leaves us with \( y = z - x \), an equation where \( y \) is expressed in terms of \( x \) and \( z \). The main steps in the isolation technique include:

• Identify the variable to be isolated.
• Perform operations that helps remove other terms from the side containing the variable.
• Ensure operations are done equally to both sides to maintain balance.

This technique is powerful because it can be applied to various types of equations, simplifying even more complex problems.