Problem 48 - A Graphical Approach to College Algebra

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\). $$\sum_{i=1}^{4}\left(4-6 x_{i}\right)$$

Step-by-Step Solution

Answer

The value of the sum is 28.

Step 1: Understand the Sum

The given sum is \( \sum_{i=1}^{4} (4 - 6x_i) \). It asks us to calculate the expression \( 4 - 6x_i \) for each value of \( x_i \) from \( x_1 \) to \( x_4 \), and then add those results together.

Step 2: Evaluate Each Term

Calculate \( 4 - 6x_i \) for each \( x_i \):1. For \( i = 1 \):\[ 4 - 6x_1 = 4 - 6(-2) = 4 + 12 = 16 \]2. For \( i = 2 \):\[ 4 - 6x_2 = 4 - 6(-1) = 4 + 6 = 10 \]3. For \( i = 3 \):\[ 4 - 6x_3 = 4 - 6(0) = 4 \]4. For \( i = 4 \):\[ 4 - 6x_4 = 4 - 6(1) = 4 - 6 = -2 \]

Step 3: Sum the Evaluated Terms

Add the evaluated terms from Step 2:\[ 16 + 10 + 4 - 2 = 28 \]

Step 4: Final Answer

The value of the sum \( \sum_{i=1}^{4} (4 - 6x_i) \) is \( 28 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations, like addition and subtraction. They are essential in algebra because they allow us to represent mathematical situations and relationships in a generalized form.

For example, in the exercise, the expression is \(4 - 6x_i\). Here, \(4\) is a constant, and \(-6x_i\) is a term that includes the variable \(x_i\) multiplied by \(-6\). Variables represent unknown quantities, which can change, depending on the problem's context.

Algebraic expressions can contain several terms. Each term is a part of the expression separated by a plus or minus sign. Understanding how to manipulate these terms is a fundamental skill that makes evaluating algebraic expressions much more straightforward.

Working with algebraic expressions helps us model real-world situations, predict outcomes, and solve problems in a flexible way.

Evaluating Sums

Evaluating sums involves adding up a series of terms. In algebra, a sum often involves variable expressions, which means calculating the result for each given value of the variable and then adding them together.

For instance, in the exercise, the sum \( \sum_{i=1}^{4} (4 - 6x_i) \) requires evaluating the expression for each \(x_i\) value from \(x_1\) to \(x_4\).

• For \(i=1\), evaluating gives \(16\).
• For \(i=2\), evaluating gives \(10\).
• For \(i=3\), evaluating gives \(4\).
• For \(i=4\), evaluating gives \(-2\).

The next step is to add these evaluated results together, leading to a total of \(28\).

This process helps us understand, analyze, and solve problems by systematically working through each part of the sum.

Mathematical Notation

Mathematical notation is a universal language used to represent mathematical concepts and relationships precisely and concisely. Understanding this notation is essential to grasp many mathematical ideas.

In the exercise, the summation notation \( \sum \) represents the addition of terms according to a specified rule. The expression, \( \sum_{i=1}^{4} (4 - 6x_i) \), instructs us to perform the operation from \(i=1\) to \(i=4\).

Each part of the notation has a specific meaning:

• \( \sum \) Sign: Indicates the sum is to be calculated.
• Subscript \(i=1\) to \(4\): Tells us the range of values for \(i\), from 1 to 4.
• Expression \(4 - 6x_i\): Shows what is calculated for each value of \(i\).

Using mathematical notation like this allows us to represent complex operations efficiently and consistently. With practice, reading and writing these notations becomes second nature, enabling deeper understanding and communication in mathematics.