Problem 4 - Elementary Algebra

Question

Evaluate for the given set of \(x\) -values. $$ x+6 x-5 ;\\{-6,0,5\\} $$

Step-by-Step Solution

Answer

For \( x = -6 \), result is \(-47\); for \( x = 0 \), result is \(-5\); for \( x = 5 \), result is \( 30 \).

Step 1: Simplify the Expression

The given expression is \( x + 6x - 5 \). First, we combine like terms. The terms \( x \) and \( 6x \) are like terms. Add them together to get \( 7x \). Thus, the simplified expression is \( 7x - 5 \).

Step 2: Evaluate for \( x = -6 \)

Substitute \( x = -6 \) into the expression \( 7x - 5 \). This gives us: \( 7(-6) - 5 = -42 - 5 \). Calculate the result: \(-42 - 5 = -47\).

Step 3: Evaluate for \( x = 0 \)

Substitute \( x = 0 \) into the expression \( 7x - 5 \). This gives us: \( 7(0) - 5 = 0 - 5 \). Calculate the result: \( 0 - 5 = -5 \).

Step 4: Evaluate for \( x = 5 \)

Substitute \( x = 5 \) into the expression \( 7x - 5 \). This gives us: \( 7(5) - 5 = 35 - 5 \). Calculate the result: \( 35 - 5 = 30 \).

Step 5: Final Step: Present Results

The evaluated results for each \( x \) value are: for \( x = -6 \), the result is \(-47\); for \( x = 0 \), the result is \(-5\); and for \( x = 5 \), the result is \( 30 \).

Key Concepts

Evaluation

In mathematics, evaluation refers to the process of finding the value of an algebraic expression by substituting variables with given numbers. This process helps us determine specific outcomes based on different inputs. For example, in the exercise provided, you are given a polynomial expression, and you need to substitute different values for the variable \( x \). This involves calculating the resulting value of the expression when \( x \) takes on different values such as \( -6 \), \( 0 \), and \( 5 \).To evaluate the expression, you do the following steps:

• Replace the variable with the given number.
• Perform the arithmetic operations following the order of operations - multiplication and division first, then addition and subtraction.
• Record the result as the evaluated value of the expression for that specific variable input.

By methodically substituting each \( x \) value, you see how the results vary, providing insights into how the expression behaves under different conditions.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that involves making an expression easier to work with by combining like terms and eliminating unnecessary components.In our example, the expression starts as \( x + 6x - 5 \). Here, you combine the terms that are similar, known as 'like terms'. Both \( x \) and \( 6x \) are like terms.

Why simplify?

• It reduces complexity, making calculations easier.
• Helps in understanding the structure and behavior of the expression.

Simplifying the initial expression yields \( 7x - 5 \), which is more straightforward. By simplifying, you prepare the expression for evaluation, making further operations quicker and less error-prone. This process involves carefully aligning terms to combine all coefficients of the same variable and resolving any constants.

Like Terms

Understanding and identifying like terms is crucial for simplifying expressions efficiently. Like terms are components of an algebraic expression that have the same variables raised to the same powers, though their coefficients can differ.For instance, in the expression \( x + 6x \), both terms are like terms because they contain the variable \( x \) raised to the first power. The coefficients, which are numbers multiplying the variable, are \( 1 \) and \( 6 \), respectively.

How to combine like terms:

• Add or subtract the coefficients directly.
• Keep the variable part unchanged.

Therefore, \( x + 6x \) simplifies to \( 7x \), as the coefficients \( 1 \) and \( 6 \) sum to \( 7 \). This reduction process is integral to making expressions manageable and prepare them for subsequent evaluation steps.