Problem 37 - Algebra 2

Question

Simplify each expression. $$ (3 q-6)-(q+13)+(-2 q+11) $$

Step-by-Step Solution

Answer

The simplified expression is \(-8\).

Step 1: Remove Parentheses

Start by removing the parentheses in the expression. We must be careful with signs when removing parentheses, especially with subtraction. We can rewrite the expression as: \( 3q - 6 - q - 13 - 2q + 11 \).

Step 2: Combine Like Terms

Now, group like terms together. Combine the terms with \( q \) and the constant terms. Gather the \( q \) terms: \( 3q - q - 2q \).Gather the constant terms:\( -6 - 13 + 11 \).

Step 3: Simplify \( q \) Terms

Simplify the terms with \( q \):\( 3q - q - 2q = 0q \).So this part of the expression simplifies to 0.

Step 4: Simplify Constant Terms

Simplify the constant terms:\( -6 - 13 + 11 = -8 \).

Step 5: Write the Simplified Expression

Combine the simplified \( q \) terms and constant terms to write the final simplified expression: \( 0q - 8 = -8 \).

Key Concepts

Removing Parentheses

When simplifying an expression, one critical step is removing parentheses. Parentheses are symbols that group parts of a mathematical expression. To remove them, you need to pay close attention to the signs in front of them. It's not just about dropping these brackets but making sure you apply the operation, especially with subtraction.

For example, if you have an expression like

• \((3q - 6) - (q + 13) + (-2q + 11)\),

- you need to distribute the negative sign before removing the parentheses.

Thus, it becomes:

• \(3q - 6 - q - 13 - 2q + 11\).

Remember, reversing the sign of each term inside the parentheses is necessary when a negative sign is in front of the bracket. This ensures accuracy in simplifying expressions. Removing parentheses helps you see what terms you can combine next.

Combining Like Terms

The next step in simplifying expressions is combining like terms. Like terms are terms that have the same variable raised to the same power. Only these terms can be combined by adding or subtracting their coefficients.

In our expression, once parentheses are removed, you have:

• \(3q - 6 - q - 13 - 2q + 11\).

Here, like terms include those containing the variable \(q\) and constant numbers. Let’s group them:

• For \(q\) terms: \(3q, -q, -2q\)

• For constants: \(-6, -13, +11\).

Combining like terms simplifies the equation, reducing the complexity and making problem-solving easier.

Distributive Property

The distributive property is essential in algebraic expressions, especially when removing parentheses or simplifying further. It's used to multiply a single term and two or more terms inside parentheses. This property states that

• \(a(b + c) = ab + ac\).

In our exercise, we utilize a similar idea when handling subtraction across parentheses. The negative sign outside acts like a multiplier, requiring each term inside to change its sign. Consider:

• \(-(q + 13)\) turns into \(-q - 13\).

This same principle is applied to all terms affected by operators outside of parentheses. Remembering this fundamental concept will help when distributing factors throughout an expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators like addition and subtraction. Simplifying these expressions requires several steps to ensure they are in their most reduced form. In our example,

• \(3q - 6 - q - 13 - 2q + 11\),

involves removing parentheses, combining like terms, and understanding properties like distribution. Live interaction with these expressions improves grasping algebraic concepts.

Algebra is about recognizing patterns and relationships between numbers and symbols. The goal when simplifying is to make expressions as straightforward as possible while retaining equality.

Mastering these processes allows for efficient and effective handling of more complex equations in the future.