Problem 69 - Algebra 2

Question

Evaluate each expression if \(a=-5, b=0.25, c=\frac{1}{2},\) and \(d=4 .(\text { lesson } 1-1)\) $$ a+2 b-c $$

Step-by-Step Solution

Answer

The expression evaluates to -5.

Step 1: Substitute the Values

Replace the variables in the expression with the given values. The expression we have is \( a + 2b - c \). Substitute \( a = -5 \), \( b = 0.25 \), and \( c = \frac{1}{2} \). This gives us \( -5 + 2(0.25) - \frac{1}{2} \).

Step 2: Simplify the Multiplication

Focus on simplifying the multiplication step in the expression. Evaluate \( 2 \times 0.25 \), which results in \( 0.5 \). Now the expression is \( -5 + 0.5 - \frac{1}{2} \).

Step 3: Perform the Addition and Subtraction

Now simplify the addition and subtraction. First, perform the addition: \( -5 + 0.5 = -4.5 \). Next, subtract \( \frac{1}{2} \): \( -4.5 - \frac{1}{2} = -5 \). The expression evaluates to \( -5 \).

Key Concepts

Variable Substitution

When working with algebraic expressions, variable substitution is a fundamental technique. It involves replacing the letters in an expression with specific numbers. This is crucial when you're asked to evaluate expressions. Take the expression given in the exercise:

• The expression is: \( a + 2b - c \)
• Given values are: \( a = -5 \), \( b = 0.25 \), \( c = \frac{1}{2} \)

To use variable substitution, replace each letter with its corresponding value. This simplifies our work and transforms the expression into numerical terms we can solve. By substituting these values, the expression becomes \(-5 + 2(0.25) - \frac{1}{2}\). This step is simple yet essential, as it sets the stage for all further calculations.

Simplification Process

The simplification process helps to make complex expressions more manageable. After substituting the values into the expression from the variable substitution step, the next goal is to simplify it. This part often involves resolving multiplication, division, addition, or subtraction in a tidy sequence. In our simplified expression \(-5 + 2(0.25) - \frac{1}{2}\), the immediate task is to resolve any multiplication or division present.

• We focus on \(2 \times 0.25\), which equals \(0.5\).

After simplifying this part, the expression is transformed further to \(-5 + 0.5 - \frac{1}{2}\). Each step in simplification helps in getting closer to the final value, making intricate expressions easier to understand and solve.

Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division are the cornerstones of simplifying and solving expressions. Once you have substituted the variables and carried out initial multiplication, it's time to complete the arithmetic operations.In the expression \(-5 + 0.5 - \frac{1}{2}\), you perform the operations left-to-right:

• First, handle the addition: \(-5 + 0.5 = -4.5\).
• Then perform the subtraction: \(-4.5 - \frac{1}{2} = -5\).

These steps apply basic arithmetic rules to deduce the final answer, which is \(-5\). Understanding each operation and its correct execution sequence is vital for accurately evaluating expressions. Mastery of these operations builds confidence in tackling more complex mathematical challenges.