Writing Equivalent Algebraic Expressions with Addition and Subtraction

Writing Equivalent Algebraic Expressions with Addition and Subtraction – Definition

When we talk about algebra, we often find different expressions that can represent the same value or idea. This concept is crucial not just in mathematics but in solving problems in everyday life where flexibility and adaptation to various representations can lead to easier solutions. An equivalent algebraic expression is an expression that, although it may look different, has the same value as another expression when the variable(s) are substituted with the same number(s).

Understanding Equivalent Algebraic Expressions – Explanation

To master the art of writing equivalent expressions, a fundamental skill in algebra, it's essential to understand not just the process of adding and subtracting expressions effectively, but also to deeply comprehend the components that make up algebraic terms and expressions. This comprehension includes recognising and distinguishing between variables, constants, and coefficients, in addition to identifying like terms. Engaging with writing numerical expressions can significantly enhance your ability to manipulate and understand these algebraic concepts.

Like Terms

Adding and Subtracting Algebraic Expressions

Adding or subtracting algebraic expressions involves combining like terms by adding or subtracting their coefficients. This method is straightforward yet powerful, allowing us to simplify complex expressions into more manageable forms.

Simplify $4x + x$

• Identify like terms: In the expression $4x + x$, both terms are like terms because they contain the same variable, $x$.
• Add the coefficients of like terms: The coefficient of $x$ in the term $x$ is $1$ (since $x$ is the same as $1x$). Therefore, when adding $4x$ and $x$, you are essentially adding $4x + 1x$.
• Perform the addition: Add the coefficients $4$ and $1$ to get $5$.
• Write the simplified expression: After adding the coefficients, the simplified expression is $5x$.

Simplify $7n + 3m - 5n + m$

• Identify like terms: In the expression $7n + 3m - 5n + m$, $7n$ and $-5n$ are like terms because they both contain the variable $n$. Similarly, $3m$ and $m$ are like terms because they both contain the variable $m$.
• Add the coefficients of like terms: The coefficients of $n$ terms are $7$ and $-5$, and for $m$ terms are $3$ and $1$ (since $m$ is the same as $1m$).
• Perform the addition: Add the coefficients of $n$ terms $7$ and $-5$ to get $2$, and add the coefficients of $m$ terms $3$ and $1$ to get $4$.
• Write the simplified expression: After adding the coefficients, the simplified expression is $2n + 4m$.

Simplify $-6w + w - k + 2k$

• Identify like terms: In the expression $-6w + w - k + 2k$, $-6w$ and $w$ are like terms because they both contain the variable $w$. Similarly, $-k$ and $2k$ are like terms because they both contain the variable $k$.
• Add the coefficients of like terms: The coefficients for $w$ terms are $-6$ and $1$ (since $w$ is the same as $1w$), and for $k$ terms are $-1$ (since $k$ is the same as $-1k$) and $2$.
• Perform the addition: Add the coefficients of $w$ terms $-6$ and $1$ to get $-5$, and add the coefficients of $k$ terms $-1$ and $2$ to get $1$.
• Write the simplified expression: After adding the coefficients, the simplified expression is $-5w + k$.

Practice combining like terms on your own now.

Writing Equivalent Algebraic Expressions – Application

Example 1: Imagine participating in a month-long fitness challenge. In the first week, you plan to complete $x$ minutes of cardio and double that amount in strength training. In the second week, you aim to increase your cardio by $30$ minutes while keeping strength training constant. How can you express the total minutes spent on each activity over the two weeks?

Expression:

• First week cardio: $x$ minutes.
• First week strength training: $2x$ minutes.
• Second week cardio: $x + 30$ minutes.
• Total minutes on cardio over two weeks: $x + (x + 30)$.
• Total minutes of strength training over two weeks: $2x$ (no change in the second week).

Combining Like Terms:

• Total cardio minutes: $2x + 30$.
• Total strength training minutes: $2x$.

Example 2: You're working on an art project that requires blue and yellow paint. On the first day, you use $b$ ounces of blue paint and $y$ ounces of yellow paint. The next day, inspired, you decide to double your usage of blue paint but use half the amount of yellow paint. How can you express the total paint used?

Expression:

• First day blue paint: $b$ ounces.
• Second day blue paint: $2b$ ounces.
• Total blue paint used: $b + 2b$.
• First day yellow paint: $y$ ounces.
• Second day yellow paint: $\frac{y}{2}$ ounces.
• Total yellow paint used: $y + \frac{y}{2}$.

Combining Like Terms:

• Total blue paint: $3b$ ounces.
• Total yellow paint: $\frac{3y}{2}$ ounces.

Example 3: Your book club decides to read $n$ pages of a novel each week. However, during the second week, the club agrees to read an additional $10$ pages to finish earlier. How can you express the total pages read after two weeks?

Expression:

• Week 1 pages: $n$.
• Week 2 pages (with additional): $n + 10$.

Combining Like Terms:

• Total pages read after two weeks: $2n + 10$.

Try some on your own!

Writing Equivalent Algebraic Expressions with Addition and Subtraction – Summary

Writing Equivalent Algebraic Expressions with Addition and Subtraction – Frequently Asked Questions