Problem 42 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-\\{6 x+3 y[-2(x+4 y)]\\}, \text { if } x=0 \text { and } y=1$$

Step-by-Step Solution

Answer

The value of the expression is 24.

Step 1: Substitute the Values

First, substitute the values of \(x = 0\) and \(y = 1\) into the given expression: \[-\{6\cdot 0 +3\cdot 1[-2(0+4\cdot 1)]\}\]This simplifies the expression to:\[-\{0 +3[-2(0+4)]\}\]

Step 2: Simplify Inside the Brackets

Next, calculate the expression inside the round brackets:\[-2(0 + 4) = -2(4) = -8\]The expression now simplifies to:\[-\{3[-8]\}\]

Step 3: Multiply the Expressions

Now multiply 3 by -8:\[3(-8) = -24\]Thus, the expression reduces to:\[-(-24)\]

Step 4: Apply the Negative Sign

Finally, apply the external negative sign:\[-(-24) = 24\]This gives us the final value of the expression.

Key Concepts

Substitution

Substitution in algebraic expressions involves replacing variables with known values. Imagine you have a recipe and you need to switch out ingredients. That's substitution in mathematics.
In the given exercise, we are asked to evaluate the expression for specific values of variables, namely, when \( x = 0 \) and \( y = 1 \).
Here’s how substitution works in this context:

• Identify the variables in your expression. In our exercise, these are \( x \) and \( y \).
• Replace each variable with the given value. Substitute \( x = 0 \) and \( y = 1 \) into the expression \(-\{6x + 3y[-2(x + 4y)]\}\).
• The expression changes to: \(-\{6\cdot 0 + 3\cdot 1[-2(0 + 4\cdot 1)]\}\).

The substitution sets the stage for further simplification, crucial for solving the expression correctly.

Simplifying Expressions

Simplifying expressions means breaking them down into their simplest form. It involves performing operations and reducing the complexity of the algebraic expression.
In our example, after substitution, the expression looks like \(-\{0 +3[-2(0+4)]\}\). Let's delve into simplifying it:

• First, perform operations inside the innermost brackets. Calculate \(-2(0 + 4)\), which simplifies to \(-8\).
• The expression now reads \(-\{3[-8]\}\). Multiply \(3\) by \(-8\), resulting in \(-24\).
• This transforms the expression to \(-(-24)\). Now, the double negatives simplify this to \(24\).

Simplification is a step-by-step process that transforms a complex expression into something more manageable and understandable.

Order of Operations

Understanding the order of operations is fundamental in accurately solving algebraic expressions. We often remember this order with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our expression, precise steps were followed based on this order:

• First, solve any operations inside the parentheses or brackets. This involved calculating \(-2(0+4)\).
• Next, handle any multiplication or division. Here, \(3(-8)\) was computed.
• Finally, apply any addition or subtraction and negations. This results in \(-(-24)\), converting to \(24\).

Understanding and applying the order of operations ensures that you arrive at correct answers and avoid common mistakes in evaluation of expressions.