Problem 93 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-(2 y-5)$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \( -2y + 5 \).

Step 1: Apply algebraic rules

Use properties of exponents, radicals, fractions, or algebraic identities to simplify the expression.

Step 2: State the result

The simplified expression is The simplified algebraic expression is \( -2y + 5 \)..

Key Concepts

Distributive Property

Understanding the distributive property is crucial when dealing with algebraic expressions. It allows us to remove parentheses by distributing a factor across terms inside the parentheses. Simply put, the distributive property tells us how to handle an expression like \(a(b+c)\), which becomes \(ab + ac\). This rule applies regardless of whether the terms inside the parentheses are being added or subtracted.

In the given exercise, we apply this rule by multiplying -1 (think of the negative sign in front of the parentheses as -1) with each term inside the parentheses: \(2y\) and -5. So, \( -(2y-5)\) becomes \( -1*2y + -1*(-5)\). The distributive property simplifies the process and helps in breaking down more complex expressions into simpler components. A key point to remember is that multiplication is distributive over addition as well as subtraction.

When multiplying a negative number, like -1, with an expression, it changes the sign of each term in the expression. Hence, \( -1*2y\) gives us \( -2y\), and \( -1*(-5)\) gives us \( +5\), leading to the simplified expression \( -2y + 5\).

Algebraic Expression Operations

Working with algebraic expressions involves various operations such as addition, subtraction, multiplication, and division. Simplifying an algebraic expression means to make it as simple as possible by performing all possible operations and combining like terms. Like terms are terms that have the same variables raised to the same power, but they might have different coefficients. For instance, \(2y\) and \(3y\) are like terms, but \(2y\) and \(2x\) are not.

Our task often begins by applying the distributive property, as seen in the original exercise. After distribution, we then combine like terms if necessary. In doing so, we should remember to keep the variable part of the term identical; only the coefficients (the numbers in front of the variables) will change according to the operation performed. It's also important to pay attention to the signs of the terms, since subtracting a negative is equivalent to adding a positive, as demonstrated in the simplified answer of our example.

Multiplication Over Addition

The concept of multiplication over addition is integral in simplifying algebraic expressions. It is a component of the distributive property, which asserts that multiplication will distribute across terms that are added together within parentheses. This essentially means that if you have a multiplication sign outside of parentheses, you must multiply the outside term by each term inside the parentheses individually.

For instance, \( 3(x + 4)\) is an example of multiplication over addition, where we multiply 3 by both \(x\) and 4, leading to \(3x + 12\). This order of operations is vital, as failing to distribute correctly can lead to incorrect solutions. Students should remember that multiplication must be performed before addition when no parentheses suggest otherwise, according to the rules of order of operations (PEMDAS/BODMAS).

It's beneficial to imagine the multiplication sign reaching over to each term inside the parentheses, ensuring that each one is multiplied by the outside factor. This visual can help prevent mistakes and ensure a thorough understanding of the process involved in simplifying algebraic expressions.