Problem 74 - College Algebra

Question

simplify each algebraic expression. $$ \frac{1}{2}(2 y)+[(-7 x)+7 x] $$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(y\).

Step 1: Applying the Distributive Property

Distribute the \(\frac{1}{2}\) into the parenthesis to multiply it by \(2y\). This action gives \(1y\) or simply \(y\). So, the expression becomes: \(y + [(-7x)+7x]\).

Step 2: Applying the Additive Inverse Property

Simplify the term inside the square brackets. Here, \(-7x\) and \(7x\) are additive inverses, so their sum is zero. This means the whole expression simplifies to: \(y + 0\).

Step 3: Final Simplification

Remove the unnecessary zero since adding zero to a term does not change its value. Therefore, the simplified expression is \(y\).

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to simplify expressions by multiplying one term across terms inside a parenthesis. It is expressed as:

• For example, if you have an expression like \(a(b + c)\), you can distribute \(a\) across both \(b\) and \(c\) to get \(ab + ac\).
• This property holds true even when fractions or decimals are involved.

In our original exercise, the distributive property was used to handle the term \(\frac{1}{2}(2y)\). By multiplying \(\frac{1}{2}\) by \(2y\), we distributed \(\frac{1}{2}\) to \(2\), resulting in \(1y\), or simply \(y\).
This step simplifies the expression by eliminating the need for the parentheses and reducing it.
The distributive property is especially useful when simplifying expressions in algebra, so getting familiar with it is crucial for solving more complex equations.

Additive Inverse

The concept of additive inverse refers to a number which, when added to the original number, results in zero. This is a handy property that helps in simplifying terms in an expression. For any number \(x\), its additive inverse is \(-x\).

• For instance, in the expression \[(-5) + 5 = 0\], \(-5\) and \(5\) are additive inverses.
• This property is used to cancel out terms in an algebraic expression, making it simpler.

In our problem, the additive inverse property is applied to \([-7x + 7x]\). Since \(-7x\) and \(7x\) are each other's additive inverses, they sum to zero.
This step streamlines the expression by removing terms that equate to zero, reducing complexity and focusing on what's left.

Expression Simplification

Expression simplification is its own little act of magic in algebra. It's the process of making an expression as compact and efficient as possible without changing its meaning. Simplifying expressions involves applying properties like distributive and additive inverse to untangle complicated algebraic expressions.

• This means changing \(2 + 3x - 0\) to \(2 + 3x\), by eliminating zero.
• Or rearranging terms to combine like terms, using properties that simplify calculations.

In our exercise, the expression \[y + 0\]\ is reduced to just \(y\) by recognizing that adding zero does not change the value of the expression.
Understanding each piece of the expression and how they interact is key.
Simplification helps in spotting the core elements, making it easier to solve or further manipulate the expression.This process not only ensures your answer is correct, but also neatly packaged as the simplest form possible, laying the groundwork for more advanced operations.