Problem 72 - Algebra 2

Question

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ 8 b-5 $$

Step-by-Step Solution

Answer

The expression evaluates to -11.

Step 1: Identify the variables

Substitute the given values for each variable in the expression. Here, the variables are \(a\), \(b\), and \(c\). For this expression, we only need the value of \(b = -\frac{3}{4}\).

Step 2: Substitute the values into the expression

Replace \(b\) with \(-\frac{3}{4}\) in the expression \(8b - 5\). The expression becomes \(8(-\frac{3}{4}) - 5\).

Step 3: Calculate the multiplication

Multiply 8 by the value of \(b\), which is \(-\frac{3}{4}\): \[8 \times -\frac{3}{4} = -6\]

Step 4: Perform the subtraction

Subtract 5 from the result of the multiplication: \[-6 - 5 = -11\]

Step 5: Write the final answer

After performing all calculations, the final answer to the expression \(8b - 5\) is \(-11\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that allows us to replace a variable with its given value. This process is like solving a puzzle, where you match the pieces by replacing the symbols with numbers. To effectively substitute:

• Identify the variables in the equation or expression.
• Look at the problem instructions to find out what values each variable should have.
• Carefully replace each variable with its corresponding number in the expression.

This method simplifies expressions, making them easier to solve. Always double-check that each substitution is correct to avoid errors later on.

Arithmetic Operations

Arithmetic operations are the backbone of solving expressions once substitution is complete. These operations include addition, subtraction, multiplication, and division. Each operation follows specific rules that you should remember to get correct results.For example:

• Multiplication: Multiply numbers as usual, keeping an eye on negative signs. A positive times a negative gives a negative result, as seen when multiplying 8 by \(-\frac{3}{4}\).
• Subtraction: Take away one number from another. Pay attention to the signs to ensure accuracy, such as subtracting 5 from -6 to get -11.

Performing arithmetic operations step by step ensures clarity and helps avoid mistakes. Always follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together form a mathematical phrase. Think of them as sentences with numbers and letters. To work with algebraic expressions effectively:

• Identify the components: Numbers (constants), variables, and the operations connecting them.
• Understand the purpose: Expressions represent calculations you need to carry out, such as "8b - 5" in the problem.
• Simplify: Use substitution and arithmetic operations to turn the expression into a single numerical result.

By mastering algebraic expressions, you can solve various problems, from simple arithmetic to more complex algebraic equations. They form the language of algebra, allowing us to describe relationships and solve real-world problems efficiently.