Problem 38 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ (7 y+7)-(y-6) $$

Step-by-Step Solution

Answer

The expression simplifies to \(6y + 13\).

Step 1: Remove Parentheses

First, distribute the negative sign across the terms in the parentheses for the expression \(y - 6\). This gives us: \((7y + 7) - y + 6\).

Step 2: Simplify the Expression

Combine like terms. Group the terms with \(y\) together and the constant terms together. This results in: \(7y - y + 7 + 6\).

Step 3: Combine Like Terms

Add the like terms: \(7y - y = 6y\) and \(7 + 6 = 13\). This gives us: \(6y + 13\).

Key Concepts

Combining Like Terms

In algebra, an essential technique to master is combining like terms. This skill helps to simplify expressions and make them easier to work with. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(7y - y + 7 + 6\), \(7y\) and \(y\) are like terms since they both have the variable \(y\). Similarly, the numbers \(7\) and \(6\) are like terms because they are constant terms, or numbers without variables.

To combine like terms, follow these steps:

• Identify the terms with the same variables. Add or subtract the coefficients (the numbers in front of the variables).
• Do the same for constant terms by directly adding or subtracting them.

Applying this to our example:

• Combine \(7y\) and \(y\): \(7y - y = 6y\).
• Finally, combine the constants \(7\) and \(6\): \(7 + 6 = 13\).

Thus, the simplified expression becomes \(6y + 13\). Combining like terms is a critical step that allows you to tidy up expressions and solve equations more efficiently.

Distributive Property

The distributive property is a powerful algebraic tool that allows you to break down expressions with parentheses. It helps distribute a term across other terms inside a parenthesis to simplify an equation. This is particularly useful when you have expressions with variables and constants that need simplifying.

The rule for the distributive property is: \[ a(b + c) = ab + ac \]This means you take the term outside the parentheses and multiply it by each term inside the parentheses.

For example, let's look at the expression \((7y + 7) - (y - 6)\):

• Notice the negative sign in front of \(y - 6\). This is like multiplying each term inside the parentheses by \(-1\).
• Distribute \(-1\) across the terms: \(-1(y) = -y\) and \(-1(-6) = +6\).

This changes the expression to \(7y + 7 - y + 6\). The distributive property simplifies expressions by removing parentheses and making it easier to rearrange terms for further simplification.

Simplifying Expressions

Simplifying expressions is about making mathematical expressions as straightforward as possible. A simplified expression is easier to understand and use in calculations. The process generally involves removing parentheses, combining like terms, and ensuring the expression has no unnecessary components.

Consider the expression \((7y + 7) - (y - 6)\), which can be simplified to \(6y + 13\):

• First, we apply the distributive property to remove the parentheses by distributing the negative sign.
• Then, we combine like terms - similar variable terms and constant terms together.

The outcome is a streamlined expression, which in our case is \(6y + 13\), where everything is combined and unnecessary parentheses are removed.

Simplifying expressions not only makes them cleaner but also prepares them for solving equations, factoring, or evaluating specific values. It's a fundamental algebraic skill that builds a foundation for solving more complex problems.