Problem 10 - Fundamentals of Mathematics

Question

For problems 10 -12, simplify each expression by combining like terms. $$ 6 y+5-2 y+1 $$

Step-by-Step Solution

Answer

The expression simplifies to \( 4y + 6 \).

Step 1: Identify Like Terms

To simplify the expression, first identify the like terms. In the expression \( 6y + 5 - 2y + 1 \), the like terms are terms with \( y \) (\( 6y \) and \( -2y \)) and the constant terms (\( 5 \) and \( 1 \)).

Step 2: Combine Coefficients of Like Terms

Combine the coefficients of the \( y \) terms: \( 6y - 2y = (6 - 2)y = 4y \). Then, combine the constant terms: \( 5 + 1 = 6 \).

Step 3: Write Simplified Expression

The simplified expression is a combination of the results from the previous step: \( 4y + 6 \).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one key step is combining like terms. This process involves grouping terms that have the same variable raised to the same power. In the expression \(6y + 5 - 2y + 1\), the terms \(6y\) and \(-2y\) are like terms because they both contain the variable \(y\). Similarly, \(5\) and \(1\) are also like terms; they are constant, meaning there are no variables attached.Combining like terms helps to consolidate the expression into a simpler form. Essentially, it allows us to add or subtract the coefficients of the like terms, reducing the number of operations. This is crucial in making the expression more manageable and easier to understand.Remember the following steps:

• Identify like terms: Look for terms with the same variables and exponents.
• Combine them by adding or subtracting their coefficients.

Using these steps, you can simplify any polynomial expression efficiently.

Coefficients

Coefficients are the numerical parts of terms that include variables. In the term \(6y\), the coefficient is \(6\). When simplifying expressions, adding and subtracting terms involves focusing on these coefficients.In the expression \(6y - 2y\), you only combine the coefficients because \(y\) is common. You calculate \(6 - 2\) to get \(4\), resulting in \(4y\). This process repeats wherever like terms are involved.Understanding coefficients helps:

• Quickly identify terms that need to be combined.
• Simplify expressions by reducing the number of terms.
• Avoid confusion between the variables and their multipliers.

By mastering coefficients, you streamline the calculation process, ensuring you handle terms precisely while simplifying expressions.

Constant Terms

Constant terms are unique in algebraic expressions because they do not contain any variables. They remain the same, regardless of what happens with the variable terms. In the expression \(6y + 5 - 2y + 1\), the constants are \(5\) and \(1\).Combining constant terms involves simple arithmetic: adding or subtracting these standalone numbers. For instance, you sum \(5 + 1 = 6\), without any need for consideration about the variable part, as they are independent numbers.Why constant terms matter:

• Providing straightforward numerical additions or subtractions.
• Giving a final simplified expression a base value when variables are absent.

Keeping an eye on constant terms ensures that the numerical accuracy of your expression is correctly maintained while simplifying algebraic expressions.