Problem 75 - Elementary Algebra

Question

Simplify. $$ 23(x-y)+x-2 y $$

Step-by-Step Solution

Answer

The simplified expression is \( 24x - 25y \).

Step 1: Distribute the Coefficient

Distribute the coefficient 23 to both terms inside the parentheses. The expression \( 23(x-y) + x - 2y \) becomes \( 23x - 23y + x - 2y \).

Step 2: Combine Like Terms for x

Identify and combine the terms with \( x \). The expression \( 23x - 23y + x - 2y \) contains the terms \( 23x \) and \( x \). Adding them together gives \( 24x \). The expression is now \( 24x - 23y - 2y \).

Step 3: Combine Like Terms for y

Combine the \( y \) terms in the expression \( 24x - 23y - 2y \). The \( y \) terms are \(-23y\) and \(-2y\). Adding them together results in \(-25y\).

Step 4: Write the Simplified Expression

After combining all like terms, the simplified expression is \( 24x - 25y \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps in simplifying expressions by eliminating parentheses. It involves spreading or "distributing" the coefficient outside the parentheses to each term within the parentheses. This makes it easier to work with algebraic expressions by breaking them down into smaller parts.

For example, in the expression \( 23(x-y) \), the number 23 is the coefficient, and it needs to be multiplied by each term inside the parentheses, both \( x \) and \( -y \). Therefore, the expression \( 23(x-y) \) simplifies to:

• \( 23 \times x \) which becomes \( 23x \)
• \( 23 \times -y \) which becomes \( -23y \)

After distributing, the expression changes from \( 23(x-y) + x - 2y \) to \( 23x - 23y + x - 2y \), making it easier to handle the next steps in simplification.

Combine Like Terms

Combining like terms is the process of merging terms in an expression that have the same variable raised to the same power. This is an essential step in simplifying algebraic expressions. It reduces complexity and results in a more compact and simpler expression.

In the expression \( 23x - 23y + x - 2y \), we first focus on the terms containing \( x \):

• \( 23x \)
• \( x \)

Adding these terms together, \( 23x + x \) gives us \( 24x \).

Next, we look at the \( y \) terms:

• \(-23y \)
• \(-2y \)

Merging these terms becomes \(-23y - 2y = -25y \). By combining like terms, you reach the final simplified form: \( 24x - 25y \). This technique not only simplifies the expression but also makes it easier to solve and understand.

Variable Expression

A variable expression consists of numbers, variables, and mathematical operations. Variables are symbols, often represented with letters like \( x \) and \( y \), that stand for unknown values in an expression. These allow you to write general mathematical statements that can hold true for a range of numbers.

For the exercise \( 23(x-y)+x-2y \), the expression includes the variables \( x \) and \( y \). The role of these variables is crucial:

• Variables provide flexibility and represent values that can change or vary.
• They allow expression generalization so that a single expression can describe many situations.
• Using variables, one can formulate algebraic equations that model real-world scenarios.

Understanding variable expressions helps develop a foundational appreciation of algebra and facilitates the solving of equations by dealing with unknown values in a structured way.

Coefficients

Coefficients are the numerical factors that multiply variables in algebraic expressions. They are essential in algebra as they define the magnitude or amount to which the variable is multiplied.

In the expression \( 23x - 23y + x - 2y \), observe the coefficients:

• \( 23 \) in \( 23x \), implying 23 times \( x \)
• \(-23 \) in \(-23y \), which means negative 23 times \( y \)
• A hidden 1 in \( x \), written as \( 1x \), indicating 1 times \( x \)
• \(-2 \) in \(-2y \), indicating negative 2 times \( y \)

Recognizing coefficients aids in distributing processes and combining terms. They help define the behavior of variable terms when expressions are solved or simplified, making them integral to understanding algebra.