Problem 38 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-(10 x-2 y+5 z) \text { if } x=2, y=8, \text { and } z=-1$$

Step-by-Step Solution

Answer

The value of the expression is 1.

Step 1: Substitute the Given Values

Start by substituting the given values for the variables into the expression: \( x = 2 \), \( y = 8 \), and \( z = -1 \). The expression becomes:\[- (10(2) - 2(8) + 5(-1)) \]

Step 2: Simplify Inside the Parentheses

Carry out the multiplication within the brackets first:- Calculate \( 10 \times 2 = 20 \)- Calculate \( 2 \times 8 = 16 \)- Calculate \( 5 \times (-1) = -5 \)Now substitute these results back into the expression:\[- (20 - 16 - 5) \]

Step 3: Evaluate the Expression inside the Parentheses

Simplify the arithmetic inside the parentheses:- Start with the subtraction: \( 20 - 16 = 4 \)- Then, subtract 5: \( 4 - 5 = -1 \)Now the expression inside the parentheses is \(-1\).

Step 4: Apply the Negative Sign

Apply the negative sign outside of the parentheses to the result:\(-(-1) = 1\).

Step 5: State the Final Result

The value of the expression \(-(10x - 2y + 5z)\) for \(x = 2\), \(y = 8\), and \(z = -1\) is 1.

Key Concepts

Substitution Method

The substitution method is a technique used to simplify an expression or equation by replacing the variables with the given values. This is especially useful in algebra whenever specific values for the variables are provided.

In this exercise, we needed to find the value of the expression \(-(10x - 2y + 5z)\) for \(x = 2\), \(y = 8\), and \(z = -1\). By substituting the given values directly into the expression, we transform it into a numerical problem:

• Replace \(x\) with 2
• Replace \(y\) with 8
• Replace \(z\) with -1

This step gives us: \(-(10(2) - 2(8) + 5(-1))\).

By substituting known values, the expression becomes easier to work with since we immediately translate it from an algebraic expression to a purely arithmetic problem. The substitution method helps eliminate guesswork and allows for precise calculations.

Order of Operations

When working with algebraic expressions, following the correct order of operations is crucial to obtain accurate results. These rules guide us in determining which operations to perform first.

The commonly used acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)—helps remember this order. In the exercise solution, we follow these steps:

• First, perform operations inside the parentheses: calculate each multiplication involving the substituted values, such as \(10 \times 2\), \(2 \times 8\), and \(5 \times (-1)\).
• Next, simplify the expression by doing the arithmetic operations inside the parentheses, following the subtraction steps.
• Finally, handle the negative sign outside the bracket.

When each step is followed carefully, keeping in line with the order of operations, the expression simplifies to the correct numerical result.

Negative Numbers

Negative numbers play a significant role in algebraic expressions, especially when they appear either as variables or constants in expressions. Understanding their behavior is crucial for accurate calculations.

In this problem, we dealt with \(z = -1\) and a negative sign before the entire expression, which led us to a key concept:

• Multiplying a positive number by a negative number results in a negative product, as seen in \(5 \times (-1) = -5\).
• Applying subtraction with a negative number is akin to adding its positive counterpart. For instance, \(4 - 5 = -1\) because you're essentially adding \(4 + (-5)\).
• Finally, when a double negative occurs, like with \(-(-1)\), they cancel each other out, yielding a positive result: thus are equal in value to 1.

Being comfortable with negative numbers and their interaction with operations like subtraction and multiplication is essential to solving algebraic expressions accurately.