Problem 76 - Elementary Algebra

Question

Simplify. $$ -13(6 x-1)+12(4 y-1)-(-2 x+2 y-16) $$

Step-by-Step Solution

Answer

The simplified expression is \(-76x + 46y + 17\).

Step 1: Expand the expressions

Begin by distributing the constant in front of each expression within the parentheses. Expand \(-13(6x - 1)\) to get \(-13 \times 6x + (-13) \times (-1) = -78x + 13\).Next, expand \(12(4y - 1)\) to get \(12 \times 4y + 12 \times (-1) = 48y - 12\).Finally, simplify the negative of the last expression \(-(-2x + 2y - 16)\) to get \(2x - 2y + 16\).

Step 2: Combine the terms

Now combine all the expanded expressions:\(-78x + 13 + 48y - 12 + 2x - 2y + 16\).

Step 3: Group like terms

Group the like terms together:\((-78x + 2x) + (48y - 2y) + (13 - 12 + 16)\).

Step 4: Simplify each group

Simplify each group of like terms:- Combine the x-terms: \(-78x + 2x = -76x\).- Combine the y-terms:\(48y - 2y = 46y\).- Combine the constant terms: \(13 - 12 + 16 = 17\).

Step 5: Write the simplified expression

The simplified expression is: \(-76x + 46y + 17\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to distribute a single term across terms within a parenthesis. This property states that for any numbers or expressions, say \( a \), \( b \), and \( c \), the expression \( a(b + c) \) can be expanded to \( ab + ac \). This is especially useful for simplifying expressions that involve multiplication over addition or subtraction.

In the problem, we applied the distributive property to three segments:

• First, expanding \(-13(6x - 1)\) involved multiplying both terms inside the parentheses by \(-13\), resulting in \(-78x + 13\).
• Second, expanding \(12(4y - 1)\) followed the same process, yielding \(48y - 12\).
• In the last segment, \(-(-2x + 2y - 16)\), we distribute the negative sign to change the signs of each term inside to get \(2x - 2y + 16\).

Understanding how to apply the distributive property correctly is crucial for expanding expressions and making algebraic manipulation easier.

Combining Like Terms

Combining like terms is a step in simplifying expressions where we add or subtract terms that have the same variable or independent components. This process helps in reducing the complexity of expressions and prepares the equation for further solving or transformation.

In our exercise, after expanding the terms using the distributive property, we looked for terms that could be combined:

• **x-terms**: Include \(-78x\) and \(2x\). By combining them, they simplify to \(-76x\).
• **y-terms**: Consist of \(48y\) and \(-2y\). Combined, these simplify to \(46y\).
• **Constant terms**: Include \(13\), \(-12\), and \(16\). When added together, they simplify to \(17\).

Grouping and combining like terms focuses on reducing the expression's parts to their most simplified form while ensuring each type of term is clearly differentiated.

Simplification of Expressions

Simplifying expressions involves applying mathematical steps to rewrite expressions in their simplest form. This process makes expressions easier to work with and often paves the way for solving equations or inequalities.

In the simplification process of our original exercise, we:

• Started with expanding the terms using the distributive property.
• Combined like terms to bring similar terms together and reduce the complexity.
• Finally, rewrote the expression in its simplest form: \(-76x + 46y + 17\).

Simplifying expressions ensures clarity and efficiency in problem-solving by producing a concise and manageable expression. Paying attention to the proper order of operations and accurately combining terms is key to effective simplification.