Problem 77 - Algebra 2

Question

Evaluate each expression if \(a=3, b=-2,\) and \(c=1.2 .(\text { lesson } 1-1)\) \(a-[b(a-c)]\)

Step-by-Step Solution

Answer

The expression evaluates to 6.6.

Step 1: Substitute the Values

Substitute the given values of the variables into the expression. Replace \(a\) with 3, \(b\) with -2, and \(c\) with 1.2 in the expression \(a - [b(a-c)]\). The expression becomes \(3 - [-2(3-1.2)]\).

Step 2: Simplify Inside the Parentheses

Calculate the expression inside the parentheses \((a-c)\). That is \(3 - 1.2 = 1.8\). Now the expression is \(3 - [-2(1.8)]\).

Step 3: Evaluate the Multiplication

Multiply \(b\) by the result from Step 2. This is \(-2 \times 1.8 = -3.6\). Now the expression is \(3 - [-3.6]\).

Step 4: Simplify the Expression

Since subtracting a negative is equivalent to addition, simplify the expression \(3 - [-3.6]\) to \(3 + 3.6\).

Step 5: Final Calculation

Complete the final addition to get the result: \(3 + 3.6 = 6.6\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables with given numbers. This is an essential first step when evaluating expressions, as it helps transform an abstract formula into a concrete calculation. To substitute the values effectively:

• Identify which variable needs to be replaced. In our example, it is critical to note that each variable in the expression: \(a = 3\), \(b = -2\), and \(c = 1.2\).
• Replace the variables with these values in the expression. For instance, with the expression \(a - [b(a-c)]\), substitute by rewriting it as \(3 - [-2(3-1.2)]\).

This step lays the foundation for further simplification and accurate results. Ensuring you substitute correctly prevents potential errors in subsequent steps.

Simplification

Simplification in algebraic expressions involves performing operations to reduce the expression to its simplest form. After substitution, it is crucial to undertake simplification to make calculations manageable.Here's how you simplify effectively:

• First, carry out operations within parentheses. This simplifies the innermost parts of the expression first, making them easier to manage. For instance, calculate \(3 - 1.2\) to get \(1.8\), thus rewriting the expression as \(3 - [-2(1.8)]\).
• Follow through with multiplication or division next. Multiply \(-2 \times 1.8\) to obtain \(-3.6\).
• Simplify further by changing signs or combining like terms. In our case, subtraction of a negative number \(3 - [-3.6]\) becomes addition \(3 + 3.6\).

These steps ensure the expression is in its simplest form, allowing you to see and calculate the final result more easily.

Order of Operations

The order of operations is critical when working with algebraic expressions, ensuring that calculations are performed correctly and consistently. The widely-known acronym PEMDAS helps remember the right sequence: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).When dealing with the expression \(a - [b(a-c)]\), the following steps are crucial:

• Parentheses: Begin by solving operations inside any parentheses. Start with \(a-c\), calculating \(3 - 1.2\).
• Multiplication: Next, handle any multiplication or division. Here, you multiply \(-2\) with the result from the prior step, giving \(-3.6\).
• Addition/Subtraction: Finally, perform any addition or subtraction remaining, such as resolving \(3 - [-3.6]\) by changing it to \(3 + 3.6\).

Following the order of operations methodically helps to avoid mistakes and achieve the correct final result in mathematical expressions. This structured approach ensures calculations are straightforward and logical.