Problem 48 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=4, c=-1,\) and \(d=3\) $$\frac{b+c}{d}$$

Step-by-Step Solution

Answer

The value of the given expression with the provided values of the variables is 1.

Step 1: Substitute the Numbers for Variables

First, replace \(a\), \(b\), \(c\), and \(d\) in the expression with their respective values, which are -2, 4, -1, and 3 respectively. Thus, the expression becomes \(\frac{4 + (-1)}{3}\).

Step 2: Simplify the Numerator

Add the numbers in the numerator, which leads to \(\frac{3}{3}\).

Step 3: Simplify the Fraction

Lastly, divide the numerator by the denominator to get the value of the expression, which results in \(\frac{3}{3} = 1\).

Key Concepts

Understanding Variable Substitution

In prealgebra, variable substitution is a fundamental skill. It involves replacing variables in an expression with given numbers. This allows us to evaluate or solve expressions. For example, in the expression \( \frac{b+c}{d} \), we need to substitute each letter with its corresponding value.
To substitute, we need to recall the numbers assigned to each variable: \( a = -2 \), \( b = 4 \), \( c = -1 \), and \( d = 3 \).
In our expression, only \( b \), \( c \), and \( d \) are relevant. So we replace \( b \) with 4, \( c \) with -1, and \( d \) with 3. Hence, our expression transforms into \( \frac{4 + (-1)}{3} \).
This is a simple yet crucial first step. It reveals the numerical values we can work with. Mastering this allows smoother progress through complex problems.

Steps to Fraction Simplification

Fraction simplification is about making a fraction as simple as possible. This means turning it into the simplest form without changing its value. Once we have substituted the variables, the expression \( \frac{4 + (-1)}{3} \) requires us to simplify.
The first step is to handle the operations in the numerator. Add the numbers: \( 4 + (-1) \) equals 3. With this, our expression becomes \( \frac{3}{3} \).
Next, we simplify the fraction itself. When the numerator and the denominator are the same, the fraction simplifies to 1. This is because any number divided by itself equals 1.

• Perform calculations in the numerator if needed.
• Divide the simplified numerator by the denominator.

Understanding how to simplify fractions helps in efficiently solving algebraic problems.

Evaluating the Numerical Expression

Evaluating a numerical expression involves calculating the exact value of an expression after performing all necessary simplifications. Once variable substitution and simplification are done, the focus moves to evaluate the expression.
For our expression \( \frac{3}{3} \), since the numerator is equal to the denominator, the evaluation results in 1. This value represents the simplest form of the original variable expression \( \frac{b+c}{d} \).
The evaluation of an expression often involves:

• Completing all arithmetic operations like addition or subtraction.
• Applying division or multiplication as needed.
• Ensuring that expressions are in their simplest numeric form after computation.

This systematic approach of evaluating expressions is crucial. It ensures that even more complex algebraic tasks become manageable. Understanding and practicing these steps is key to mastering prealgebra.