Problem 30 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-7(2 x-3 y)+9(3 x+y)$$

Step-by-Step Solution

Answer

The simplified expression is \(13x + 30y\).

Step 1: Distribute the coefficients

Apply the distributive property to remove the parentheses from each term in the expression. This means multiplying through by the factor outside each set of parentheses: For the first term: \(-7(2x - 3y) = -7 \times 2x + (-7) \times (-3y) = -14x + 21y\)For the second term: \(9(3x + y) = 9 \times 3x + 9 \times y = 27x + 9y\)

Step 2: Combine like terms

Now that we have removed the parentheses, we can combine like terms. Like terms are terms that have the same variable raised to the same power:Collect and combine the terms with \(x\):\(-14x + 27x = 13x\)Collect and combine the terms with \(y\):\(21y + 9y = 30y\)

Step 3: Write the simplified expression

Now that we have combined the like terms, write the final simplified expression:The expression simplifies to:\(13x + 30y\)

Key Concepts

Distributive Property

When working with algebraic expressions, the distributive property is an essential tool. It allows us to eliminate parentheses and simplify expressions. This property states that when you have an expression where a term outside a parenthesis needs to be multiplied with terms inside, you multiply the outside term with each term within the parenthesis.

In mathematical terms, the distributive property is represented as:

• If you have an expression of the form \(a(b + c)\), it becomes \(ab + ac\).

In the given exercise, we applied the distributive property to remove parentheses. For example:

• For the part \(-7(2x - 3y)\), we performed \(-7 imes 2x\) and \(-7 imes (-3y)\), resulting in \(-14x + 21y\).
• Similarly, for \(9(3x + y)\), we calculated \(9 imes 3x\) and \(9 imes y\), leading to \(27x + 9y\).

Practicing this property is key as it forms the foundation for simplifying expressions and solving equations in algebra.

Combining Like Terms

After using the distributive property to eliminate parentheses, the next step in simplifying algebraic expressions is combining like terms. Like terms are components of an expression that have identical variable parts. This means they share the same variable and exponent.

In the exercise, once the parentheses are removed, we have several terms:

• \(-14x, 21y, 27x,\) and \(9y\).

To simplify:

• Combine terms with the variable \(x\): \(-14x + 27x\), which simplifies to \(13x\).
• Combine terms with the variable \(y\): \(21y + 9y\), which results in \(30y\).

The process of combining like terms helps in making the expression less complex and more manageable. It's like grouping similar items together to simplify your collection.

Algebraic Manipulation

Algebraic manipulation encompasses a variety of techniques used to rearrange and simplify algebraic expressions. It's the art of transforming expressions into different forms to make them easier to work with or solve.

This process often involves:

• Applying the distributive property to eliminate parentheses.
• Combining like terms to consolidate and simplify the expression.
• Ordering terms in a standard form, sometimes ascending or descending according to the degree or alphabetical order of the variables.

In our solution, we successfully manipulated the expression by first distributing coefficients and then combining terms with similar variables, simplifying it to \(13x + 30y\). Through algebraic manipulation, we turn complex, unwieldy expressions into more understandable and usable forms. Mastering these skills is essential to excel in algebra and solve more complex mathematical problems effectively.