Problem 73 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ \frac{2}{5} b+1 $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{7}{10} \).

Step 1: Substitute the Given Values

We are given the expression \( \frac{2}{5} b + 1 \). Substitute \(b = -\frac{3}{4}\) into the expression. The expression becomes \( \frac{2}{5} \left(-\frac{3}{4}\right) + 1 \).

Step 2: Simplify the Multiplication

Multiply \( \frac{2}{5} \) by \( -\frac{3}{4} \). This is done by multiplying the numerators and denominators separately: \[ \frac{2 \cdot (-3)}{5 \cdot 4} = \frac{-6}{20}. \] Simplify \( \frac{-6}{20} \) by dividing the numerator and the denominator by their greatest common divisor, which is 2, giving \( \frac{-3}{10} \).

Step 3: Add to the Constant Term

Add \( \frac{-3}{10} \) to \( 1 \). To do this, convert 1 into a fraction with a denominator of 10: \( 1 = \frac{10}{10} \). Now add the fractions: \[ \frac{10}{10} + \frac{-3}{10} = \frac{10 - 3}{10} = \frac{7}{10}. \]

Step 4: Write the Final Answer

The simplified value of the expression \( \frac{2}{5} b + 1 \) when \( b = -\frac{3}{4} \) is \( \frac{7}{10} \).

Key Concepts

Substitution

Substitution is a fundamental concept used in mathematics to replace variables with known values. This allows you to evaluate expressions easily. In our exercise, we see an expression that involves the variable \( b \). The first step requires substituting \( b \) with the given value of \(-\frac{3}{4}\). Here’s how it works:

• Identify the variable: In this case, the variable is \( b \).
• Find the given value of the variable: \( b = -\frac{3}{4}\).
• Replace the variable with its value in the expression: \( \frac{2}{5} b + 1 \) becomes \( \frac{2}{5}(-\frac{3}{4}) + 1 \).

Substitution helps manage expressions by eliminating variables, allowing for straightforward calculations. It's a simple yet powerful tool in algebra.

Fraction Multiplication

When multiplying fractions, you perform the operation differently than when multiplying whole numbers. Each fraction consists of a numerator and a denominator, and both are critical in multiplication. To multiply two fractions, follow these steps:

• Multiply the numerators: In our example, you have \( \frac{2}{5} \) and \( -\frac{3}{4} \). Multiply the numerators \( 2 \) and \(-3 \), resulting in \(-6\).
• Multiply the denominators: Multiply the denominators \( 5 \) and \( 4 \), resulting in \(20\).
• Combine your results: The resultant fraction is \( \frac{-6}{20} \).

Remember, when multiplying fractions, the sign of the product is determined by the rules of multiplication regarding positive and negative numbers.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, making them easier to understand and compare. The fraction \( \frac{-6}{20} \) can be simplified using this method:- Find the greatest common divisor (GCD) of the numerator and the denominator; in this case, the GCD is \(2\).- Divide both the numerator and the denominator by their GCD: Divide \(-6\) and \(20\) by \(2\). Result: \(\frac{-3}{10}\). Simplification reduces fractions to their most basic terms, revealing their true proportions or values in the simplest form. It’s a key part of ensuring mathematical expressions remain clean and intuitive.

Addition of Fractions

Adding fractions, especially when they have different denominators, requires a bit of work to make them compatible. This is done by converting each fraction to have a common denominator. Here's how:- Find a common denominator: In the expression \( \frac{-3}{10} \) and \( 1 \), converting \(1\) into a fraction with a denominator of \(10\) results in \( \frac{10}{10} \).- Add the fractions: Simply add the numerators \(10\) and \(-3\) while keeping the common denominator, leading to \( 7/10\).Remember:

• Fractions add directly across the numerators when their denominators match.
• Ensure simplified forms for clarity and ease in interpretation.

Adding fractions enables you to combine values into a single expression, providing a unified representation of mathematical quantities.