Problem 37 - Algebra 2

Question

Simplify each expression. $$ 3 x+5 y+7 x-3 y $$

Step-by-Step Solution

Answer

The expression simplifies to \(10x + 2y\).

Step 1: Identify like terms

In the expression \(3x + 5y + 7x - 3y\), identify the like terms. Like terms have the same variable raised to the same power. Here, \(3x\) and \(7x\) are like terms since both contain the variable \(x\). Similarly, \(5y\) and \(-3y\) are like terms as they both contain the variable \(y\).

Step 2: Combine like terms for x

Add the coefficients of the like terms with \(x\). The terms are \(3x + 7x\). Add \(3\) and \(7\): \(3x + 7x = 10x\). So, these terms simplify to \(10x\).

Step 3: Combine like terms for y

Add the coefficients of the like terms with \(y\). The terms are \(5y - 3y\). Add \(5\) and \(-3\): \(5y - 3y = 2y\). So, these terms simplify to \(2y\).

Step 4: Write the simplified expression

Combine the simplified terms from steps 2 and 3. The simplified expression is \(10x + 2y\), representing the addition of the results from each like term.

Key Concepts

Like Terms

When dealing with algebraic expressions, understanding the concept of like terms is crucial. Like terms are terms in an expression that have identical variables raised to the same power. This means that the only difference between the like terms can be their coefficients, or numerical values, in front of the variables.
For example, in the expression \(3x + 7x - 5y\), the terms \(3x\) and \(7x\) are like terms because they both involve the variable \(x\). Similarly, if we had another term like \(-5y\), it would be grouped with any terms involving \(y\) only, such as \(2y\).
Identifying like terms sets the stage for simplifying complex expressions, making it easier to work with and understand them in the context of solving algebraic problems.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. It involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged. This process helps in reducing the expression to its simplest form.
To combine like terms, follow these steps:

• Identify the terms in the expression that have the same variable and the same power.
• Focus on the coefficients of these terms.
• Add or subtract the coefficients to produce a single term for each group of like terms.

For the expression \(3x + 5y + 7x - 3y\), combining like terms involves adding \(3x\) and \(7x\) to get \(10x\), and \(5y\) and \(-3y\) to get \(2y\). The result is a simplified expression \(10x + 2y\).
This process saves time and simplifies solving by reducing the number of terms you need to work with.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These are the building blocks of algebraic equations and functions, allowing us to express and solve real-world problems.
In the expression \(3x + 5y + 7x - 3y\), each part like \(3x\) or \(-3y\) consists of a coefficient (a number) and a variable (a letter), which represent unknown values.
Understanding and simplifying algebraic expressions is crucial because:

• It helps in solving equations more efficiently by reducing complex expressions.
• It allows clearer communication of mathematical ideas and relationships.
• It lays the groundwork for more advanced mathematical concepts.

Learning to simplify these expressions through techniques like combining like terms ensures that you can handle more complicated algebraic problems with ease.