Problem 19 - Algebra 2

Question

Evaluate each expression if \(a=3, b=0.3, c=\frac{1}{3},\) and \(d=-1\). \(\frac{d+4}{a^{2}+3}\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{4}\).

Step 1: Substitute the values into the expression

First, substitute the given values of the variables into the expression. The expression is \( \frac{d+4}{a^{2}+3} \). Substitute \( a = 3 \) and \( d = -1 \) into the expression: \[ \frac{-1 + 4}{3^{2} + 3} \].

Step 2: Simplify the numerator

Simplify the numerator of the fraction. Compute \( -1 + 4 \): \[ -1 + 4 = 3 \]. So the numerator becomes 3.

Step 3: Calculate the denominator

Simplify the denominator of the fraction. Start by calculating \( a^{2} \): \( 3^{2} = 9 \). Then add 3: \[ 9 + 3 = 12 \]. So the denominator becomes 12.

Step 4: Simplify the whole expression

Now that you have the simplified numerator and denominator, compute the fraction: \[ \frac{3}{12} = \frac{1}{4} \].

Key Concepts

Substitution

Substitution is an essential mathematical technique used to evaluate expressions by replacing variables with their given numerical values. This method simplifies solving problems, making them more manageable and straightforward. For the expression \( \frac{d+4}{a^{2}+3} \), we start by identifying the given values: \( a = 3 \) and \( d = -1 \). By substituting these values directly into the expression, we transform it into a numeric form that is easier to handle:

• Replace \( d \) with \(-1\): \(d + 4\) becomes \(-1 + 4\)
• Replace \( a \) with \( 3 \): \(a^2 + 3\) becomes \( 3^2 + 3\)

These substitutions simplify the process of evaluating expressions by focusing only on arithmetic calculations rather than dealing with abstract variables. Remember, substitution is your first step in tackling expressions involving variables.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form so that it is easier to understand and compare. When we have a fraction like \( \frac{3}{12} \), simplifying it means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.To simplify \( \frac{3}{12} \), identify the GCD of 3 and 12:

• The factors of 3 are: 1, 3
• The factors of 12 are: 1, 2, 3, 4, 6, 12

The highest common factor between them is 3. Thus, divide both the numerator and the denominator by 3:

• Numerator: \( 3 \div 3 = 1 \)
• Denominator: \( 12 \div 3 = 4 \)

This gives us the simplified fraction \( \frac{1}{4} \). Simplification makes calculations with fractions easier and helps in clearly understanding the proportion or value they represent.

Numerator and Denominator

When dealing with fractions, it's vital to understand the roles of numerators and denominators. The fraction \( \frac{d+4}{a^{2}+3} \) is an example, where:

• The numerator is the top part (\(d+4\)), representing how many parts of the whole are being considered.
• The denominator is the bottom part (\(a^2+3\)), indicating the total number of equal parts the whole is divided into.

In our specific example, after substitution:

• The numerator becomes \( -1 + 4 = 3 \), which tells us we are considering 3 parts.
• The denominator becomes \( 3^2 + 3 = 12 \), signifying the whole is divided into 12 equal parts.

Understanding these concepts is crucial for simplifying fractions, comparing them, and performing operations such as addition, subtraction, multiplication, and division of fractions.