Problem 45 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(7 a-2 b-9 a+3 b, \quad a=4\) and \(b=-6\)

Step-by-Step Solution

Answer

The expression evaluates to -14.

Step 1: Substitute Values

Substitute the given values of the variables into the algebraic expression. Here, substitute \(a = 4\) and \(b = -6\) into the expression \(7a - 2b - 9a + 3b\). This gives us:\[ 7(4) - 2(-6) - 9(4) + 3(-6) \]

Step 2: Simplify Terms

Perform the multiplication operation for each term separately. We get:\[ 28 + 12 - 36 - 18 \]

Step 3: Combine Like Terms

Now combine all the terms by adding or subtracting as needed. The expression \(28 + 12 - 36 - 18\) simplifies as follows:1. Add \(28 + 12 = 40\).2. Add \(-36 - 18 = -54\).3. Finally, combine them: \(40 - 54 = -14\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. They express one or more terms. In simpler words, a term is either a single number or a variable, or numbers and variables multiplied together.
Terms are separated by addition or subtraction. For example, in the expression \(7a - 2b - 9a + 3b\), each element like \(7a\) and \(-2b\) is a term.

• Think of algebraic expressions as a way to represent real-life situations with letters and numbers.
• They can be as simple as \(x + 1\) or as complex as \(3x^2 - 5x + 7\).

Understanding algebraic expressions is crucial as it is the foundation of algebra, helping you solve equations by performing arithmetic operations.

Variable Substitution

Variable substitution is a fundamental concept in algebra where specific values replace variables. This process transforms an algebraic expression into a numeric expression. In our example, given the expression \(7a - 2b - 9a + 3b\) and the values \(a = 4\) and \(b = -6\), we substitute:

• Replace \(a\) with \(4\).
• Replace \(b\) with \(-6\).

This turns our expression into numeric form: \(7(4) - 2(-6) - 9(4) + 3(-6)\).
Substitution allows easier calculation and is crucial in evaluating expressions, making complex algebra more manageable.

Simplification Steps

Simplifying an expression involves performing arithmetic to reduce it to its simplest form. After substituting, the next steps are usually multiplications and then addition or subtraction of numbers. For \(7(4) - 2(-6) - 9(4) + 3(-6)\), follow these steps:

• Multiply each term: \(7 \times 4 = 28\), \(-2 \times -6 = 12\), \(-9 \times 4 = -36\), and \(3 \times -6 = -18\).
• The resulting expression is \(28 + 12 - 36 - 18\).
• Combine the terms: \(28 + 12 = 40\), then \(-36 - 18 = -54\).
• Finally, \(40 - 54 = -14\).

This simplification helps in quickly finding the value of an expression, allowing us to focus on understanding the underlying mathematics.