Problem 70 - Elementary and Intermediate Algebra

Question

Evaluate. $$ 50 \div 2 \cdot t, \text { for } t=-5 $$

Step-by-Step Solution

Answer

-125

Step 1: Identify the given values

We are given the expression \[ 50 \div 2 \cdot t \] and we need to evaluate it for \( t = -5 \).

Step 2: Substitute the value of t

Replace \( t \) with \( -5 \) in the expression. It becomes: \[ 50 \div 2 \cdot (-5) \].

Step 3: Perform the division

First, divide \( 50 \div 2 \). This gives \( 25 \). So the expression simplifies to: \[ 25 \cdot (-5) \].

Step 4: Perform the multiplication

Now multiply \( 25 \) by \( -5 \). This gives \( -125 \).

Key Concepts

Substitution in algebra

Substitution in algebra is a method where we replace a variable with a given number. This helps simplify the expression so we can solve it. In our example, we started with the expression \ 50 \div \ 2 \cdot t \ and were given that \( t = -5 \). To substitute, we replace every occurrence of the variable with the given value: \( t \) becomes \( -5 \). So our expression changes to \[ 50 \div \ 2 \cdot (-5) \]. Remember, brackets help clarify which operations to prioritize when multiple steps are involved.

Order of operations

The order of operations in mathematics tells us which calculations to perform first. It’s often remembered by the acronym PEMDAS which stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

When evaluating \( 50 \div\underline{\phantom{xxx}} 2 \cdot \ (-5) \), we follow these rules. First, perform the division: \( 50 \div \ 2 = 25 \). Then, perform the multiplication: \[ 25 \cdot (-5) = -125 \]. Following this correct order ensures we get the right answer every time.

Multiplication and division

Multiplication and division are fundamental operations in algebra. They help determine quantities based on the given numbers. When we’re given an expression with both multiplication and division, like \( 50 \div \ 2 \cdot t \), we perform these operations from left to right. Starting with division: \( 50 \div\underline{\phantom{xxx}} 2 \equals \ 25 \), next, multiply this result by \( t \), which in our case, is \( -5 \). Multiplying \( 25 \ by \ (-5) \) gives us \( -125 \). This left-to-right process is crucial, as reversing the order can lead to incorrect solutions.