Problem 20 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of the expression is 7.

Step 1: Substitute the Variables

We start by substituting the given values into the expression \( \frac{1-b}{3c-3b} \). Given, \( b = 0.3 \) and \( c = \frac{1}{3} \). Therefore, the expression becomes: \[ \frac{1-0.3}{3\left(\frac{1}{3}\right)-3(0.3)}. \]

Step 2: Simplify the Numerator

The expression in the numerator \( 1 - 0.3 \) simplifies to \( 0.7 \). Thus, our expression is now: \[ \frac{0.7}{3\left(\frac{1}{3}\right) - 3(0.3)}. \]

Step 3: Simplify the Denominator

Now, let's simplify the denominator: \( 3 \times \frac{1}{3} = 1 \) and \( 3 \times 0.3 = 0.9 \). So, the denominator simplifies to \( 1 - 0.9 = 0.1 \). The expression is now: \[ \frac{0.7}{0.1}. \]

Step 4: Evaluate the Final Expression

Now divide the simplified numerator by the simplified denominator: \( \frac{0.7}{0.1} = 7 \).

Key Concepts

Substitution

Substitution is a core concept in algebra that involves replacing variables with their actual values. In our exercise, we were given specific numeric values for variables. For instance:

• Variable \(b\) was given as 0.3.
• Variable \(c\) was provided as \(\frac{1}{3}\).

The goal is to correctly replace the variable in the expression with these values. This step is crucial because it simplifies the problem by transforming it from an abstract form to a more calculable one. After substitution, the original expression \(\frac{1-b}{3c-3b}\) becomes \(\frac{1-0.3}{3\left(\frac{1}{3}\right)-3\cdot0.3}\). Doing this immediately sets the stage for the next steps, where simplification eases the calculation process.

Simplification

Simplification in algebra means reducing expressions to their simplest form. After substituting variables, the next step is to simplify the resulting numeric expression. This involves performing all possible arithmetic operations:

• First, tackle the numerator: \(1 - 0.3 = 0.7\).
• Then, move to the denominator: calculate each term separately, \(3 \times \frac{1}{3} = 1\) and \(3 \times 0.3 = 0.9\), then subtract these to get the simplified denominator \(1 - 0.9 = 0.1\).

Simplifying both parts as much as possible is necessary for accurate results in further calculations. By reducing the complexity of the problem, it becomes easier to understand and solve, and it helps prevent errors during computation.

Numerator and Denominator

The numerator and denominator are essential parts of a rational expression. These terms describe the components of fractions:

• The numerator is the top part, representing how many parts of the whole are considered.
• The denominator is the bottom part, indicating the total number of equal parts in the whole.

In our exercise, we needed to address both separately:- The expression's numerator is \(0.7\), so it shows the part of the fraction we have.- The denominator is \(0.1\), which marks the amount of equal parts into which the whole is divided.Understanding these components as separate entities clarifies the fraction and simplifies solving the equation since the final solution involves dividing these numbers.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are algebraic expressions. They require careful handling:

• Simplify using arithmetic rules and laws of algebra.
• Consider both positive and negative signs during operations.

In our example, after substitution and simplification, the rational expression \(\frac{0.7}{0.1}\) represents our starting variable expression in a simple numerical form. Dividing the numerator by the denominator results in a simplified whole number: \(\frac{0.7}{0.1} = 7\). This demonstrates how rational expressions can be manipulated to find meaningful solutions to algebraic problems, teaching students the importance of ease and precision during calculations.