Problem 36 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ (2 y+20)+(5 y-30) $$

Step-by-Step Solution

Answer

The simplified expression is \(7y - 10\).

Step 1: Identify Like Terms

First, we need to identify the like terms in the expression. Like terms are terms that have the same variable raised to the same power. In the expression \((2y + 20) + (5y - 30)\), the like terms are \(2y\) and \(5y\) since they both have the variable \(y\).

Step 2: Rearrange and Combine Like Terms

Now, rearrange the expression to group the like terms together: \(2y + 5y + 20 - 30\). We will combine the terms with the variable \(y\) and the constants separately.

Step 3: Add/Subtract the Coefficients of Like Terms

Add the coefficients of the \(y\) terms: \(2 + 5 = 7\). Thus, \(2y + 5y = 7y\). For the constant terms, calculate \(20 - 30 = -10\).

Step 4: Write the Final Expression

Combine the results from Step 3 to get the simplified expression: \(7y - 10\).

Key Concepts

Like Terms

To successfully solve for polynomial expressions like \((2y + 20) + (5y - 30)\), understanding 'like terms' is crucial. Like terms are components of the expression that have identical variable parts. These parts must not only share the same variable, for instance, "y," but they must also be raised to the same power. In our exercise, the like terms are \(2y\) and \(5y\). They both involve the variable \(y\) raised to the power of one. Identifying these terms is the first step toward simplifying the expression.

• Look for terms that have the exact same variable(s) and power(s).
• Combine these like terms by performing addition or subtraction on their coefficients.

Recognizing like terms makes simplifying expressions a manageable task.

Coefficients

Coefficients are the numerical part of a term in an algebraic expression. They act as multipliers for the variables they accompany. For example, in the term \(2y\), 2 is the coefficient. In our given expression, you'll see the coefficients 2 in \(2y\) and 5 in \(5y\). Understanding coefficients allows us to add or subtract terms effectively. When combining like terms, you add or subtract their coefficients to simplify the expression.

• Only the coefficients of like terms can be added or subtracted.
• The variable and its power remain unchanged.

So, we added the coefficients of \(2y\) and \(5y\) to yield \(7y\). Remember, coefficients play a crucial role in the process of simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This involves combining like terms and calculating numerical values where possible. In our exercise, the process of simplifying starts by rearranging the expression to focus on like terms: \(2y + 5y + 20 - 30\). You then find the sum of the coefficients of the like terms, which gives \(7y\). Next, calculate any constant values, such as \(20 - 30\), resulting in \(-10\). The final step is to combine these simplified results into one expression: \(7y - 10\).

• Ensure all like terms are combined.
• Calculate constant numerical values separately.

This process aids in making an expression easier to understand and use for further operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that consist of numbers, variables, and arithmetic operations. In our case, the expression is \((2y + 20) + (5y - 30)\). Each of these components plays a role in ultimately solving or manipulating the expression comfortably. While many algebraic expressions include different elements, the primary components include:

• Variables like \(y\), standing in for unknown values.
• Coefficients, which are numbers attached to variables like 2 and 5.
• Constants, these are standalone numbers like 20 and -30.

Getting familiar with these components helps in breaking down complex algebraic expressions into simpler forms. Understanding these basics of algebraic expressions is foundational for successfully handling more intricate mathematical problems in the future.