Problem 93 - College Algebra

Question

Simplify algebraic expression. \(7-4[3-(4 y-5)]\)

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression \(7-4[3-(4 y-5)]\) is \(-16y + 39\).

Step 1- Expand and simplify expression within innermost bracket

Begin with the innermost bracket, which is \(4y - 5\). There are no operations to be applied in here, so it remains as is.

Step 2- Operate and simplify expression with outer bracket

The next expression to be handled is the next integral bracket, i.e., \(3 - (4y - 5)\). Removing the outer bracket but keeping in mind the negative sign, the equation because \(3 - 4y + 5\). Adding the constants, it simplifies to \(-4y + 8\)

Step 3- Multiply and simplify the entire equation

Multiply the result from Step 2 with the number before the bracket, which in this case is -4. So, the expression becomes \(-4*(-4y + 8)\). By applying distribution, this simplifies to \(-4*-4*y + -4*8 = 16y - 32\)

Step 4- Include the final result in the original equation and simplify

Now apply this result back to the original equation, substituting \(-4*(3 - (4y-5))\) with \(16y - 32\). This makes the equation \(7 - (16y - 32)\). Remove the brackets, this simplifies to \(7 - 16y + 32\). Reordering the terms makes the equation: \(-16y + 7 + 32\). Adding the constants gives the final simplified algebraic expression: \(-16y + 39\)

Key Concepts

Distributive Property

The distributive property is an essential algebraic principle that enables us to simplify expressions, especially when dealing with parentheses. This property states that a term can be distributed across terms inside a bracket.
For example, if you have an expression like \(a(b + c)\), you can distribute \(a\) by multiplying it with each term inside the brackets: \(a \cdot b + a \cdot c\).
In the original exercise, during step 3, the distributive property is used: \(-4(-4y + 8)\). By distributing \(-4\) across \(-4y\) and \(+8\), it simplifies to \((16y - 32)\).
This property is very helpful when you want to eliminate parentheses and deal with the expression inside directly. Remember, always be mindful of the signs (positive or negative) when applying the distribution, as they can change the values of the resulting expression.

Combining Like Terms

Combining like terms is a simplifying process where you group terms that have the same variable parts. This helps in reducing the expression to its simplest form.
Consider the expression \(-16y + 7 + 32\). Here, \-16y\ is a term with the same variable, while \+7\ and \+32\ are constants.
To combine, you simply add or subtract these terms:

• Combine the constants: \7 + 32 = 39\.
• The variable term, \(-16y\), remains as it is because there is no other \y\ term to combine it with.

The simplified expression then becomes \-16y + 39\.
It's vital to have a sharp eye when identifying like terms. They must share the exact same variables raised to the same powers to be combined. This skill is crucial for simplifying complex algebraic expressions efficiently.

Order of Operations

The order of operations is crucial when simplifying algebraic expressions to ensure each calculation is done correctly.
The conventional mnemonic "PEMDAS" helps remember this order:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

In the given exercise, the order of operations is meticulously followed:

• The expression inside the innermost parentheses, \(4y - 5\), is addressed first, even though it needs no simplification initially.
• Then, the expression \(3 - (4y - 5)\) is simplified after considering the impact of the negative sign from the subtraction.
• Finally, multiplication using the distributive property occurs before combining the resulting terms.

Always follow the order of operations to avoid errors and arrive at the correct simplified expression. It helps maintain a clear path to solving problems methodically and accurately.