Problem 38 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 2(6 x-1)-(x-7) $$

Step-by-Step Solution

Answer

The simplified expression is \(11x + 5\).

Step 1: Distribute into the first part

Begin by distributing the number 2 into the expression inside the first set of parentheses: \(2(6x - 1)\). Multiply 2 by each term inside the parentheses: \(2 \times 6x\) and \(2 \times -1\). This results in \(12x - 2\).

Step 2: Distribute into the second part

Look at the second part of the expression: \(-(x - 7)\). Distribute the negative sign (or -1) into the expression: \(-1 \times x\) and \(-1 \times -7\). This results in \(-x + 7\).

Step 3: Combine the expression

Now, combine the results from Step 1 and Step 2. You have \(12x - 2\) and \(-x + 7\). The expression becomes \(12x - 2 - x + 7\).

Step 4: Simplify the expression

Combine like terms. Combine \(12x\) and \(-x\) to get \(11x\). Then, combine \(-2\) and \(+7\) to get \(+5\). The simplified expression is \(11x + 5\).

Key Concepts

Distributive Property

When dealing with algebraic expressions, the distributive property is essential for simplifying equations and removing parentheses. It allows you to multiply a single term with each term inside a group, or parentheses. For example, in the expression \( 2(6x - 1) \), the distributive property tells us to multiply \( 2 \) by both \( 6x \) and \( -1 \). This results in \( 12x - 2 \).
Another important aspect of the distributive property is applying it to negative signs. When you come across expressions like \(-(x-7)\), it means you need to distribute a \(-1\). Multiply \(-1\) by both \( x \) and \( -7 \), resulting in \(-x + 7\).

• Multiply constants with each element in the parentheses.
• Expand the expressions while respecting operation signs.
• Ensure every term is simplified to avoid mistakes.

This process helps in breaking down complex expressions into simpler parts by using simple multiplication.

Simplifying Expressions

Simplifying an expression involves rewriting it in its simplest form, cutting down to basics without losing value. After using the distributive property to remove parentheses, the next step is to clean up the expression. This might involve rearranging terms or performing basic arithmetic operations.
Consider an expression like \( 12x - 2 - x + 7 \). To simplify, first ensure that all terms are arranged properly; this is already done here. Next, focus on operations that can make the expression shorter and more straightforward.

• Keep an eye out for similar terms that can be combined.
• Ensure numbers and coefficients are fully calculated and combined where possible.
• Perform arithmetic operations accurately to avoid errors.

Simplifying can often make complex algebra problems much easier and sometimes opens up the solution through a clearer path.

Combining Like Terms

After distributing and simplifying, combining like terms is the key to finalizing your expression. Like terms are terms that have the same variables raised to the same power. In the expression used earlier, \( 12x - 2 - x + 7 \), \( 12x \) and \(-x\) are like terms because they both contain the variable \( x \).
Combining these terms involves doing simple arithmetic on their coefficients. For \( 12x - x \), subtract 1 from 12, resulting in \( 11x \). Similarly, numerical terms like \(-2\) and \(+7\) are combined through addition or subtraction, which in this case gives \(+5\). This makes our simplified expression \( 11x + 5 \).

• Identify terms containing the same variables.
• Perform addition or subtraction directly on the coefficients.
• Ensure each unique term is simplified fully.

Mastering this step ensures you can simplify complex equations with ease, making algebra more approachable and less daunting.