Problem 51 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(-2(x+4)-(2 x-1), x=-3\)

Step-by-Step Solution

Answer

5

Step 1: Substitute the Value of x

Insert the provided value of \( x = -3 \) into the algebraic expression. The expression becomes \(-2((-3)+4)-(2(-3)-1)\).

Step 2: Simplify Inside the Parentheses

Calculate the expressions inside the parentheses. \((-3)+4 = 1\) and \(2(-3) - 1 = -6 - 1 = -7\). So the expression updates to \(-2(1) - (-7)\).

Step 3: Apply the Operations

Multiply \(-2\) by \(1\) to get \(-2\). The expression now is \(-2 - (-7)\).

Step 4: Simplify the Subtraction

Subtracting a negative number is the same as addition. Thus, \(-2 - (-7) = -2 + 7 = 5\).

Step 5: Conclude the Result

The value of the algebraic expression \(-2(x+4)-(2x-1)\) given \(x = -3\) is \(5\).

Key Concepts

Substitution in Algebra

Substitution is a foundational concept in algebra that involves replacing variables with their known values. When provided with an expression and specific values for the variables within, such as in our exercise, substitution allows us to evaluate the expression numerically. Here’s how it works:

• Identify the variable(s) within the algebraic expression. In our example, we’re working with the variable, \( x \).
• Replace each instance of the variable in the expression with its given value. For our exercise, we substitute \( x = -3 \) into the expression, transforming it into numerical form.
• Write the expression with the new substituted value, so \(-2(x+4)-(2x-1)\) becomes \(-2((-3)+4)-(2(-3)-1)\).

Substitution is critical because it converts abstract algebraic forms into concrete numbers we can work with, providing a pathway to the final solution.

Simplifying Expressions

Simplifying an expression is the process of making it easier to understand and work with. This involves combining like terms and performing arithmetic operations. During simplification:

• Start by solving inside the parentheses or brackets as these have higher precedence. For example, in our expression, the parentheses \((-3) + 4\) and \(2(-3) - 1\) are resolved first.
• Calculate these simpler, inner expressions: \((-3)+4 \rightarrow 1\) and \(2(-3)-1 \rightarrow -7\). This step is crucial in breaking down the expression into more manageable pieces.

Simplification helps eliminate unnecessary complexity, allowing focus on completing the rest of the operations efficiently.

Order of Operations

The order of operations is a rule that clarifies which procedures should be performed first in a mathematical expression to ensure consistency. Often summarized by the acronym PEMDAS:

• Parentheses – Solve expressions inside parentheses or brackets first.
• Exponents – We don’t have any in our example, but they come next.
• Multiplication and Division – Proceed from left to right.
• Addition and Subtraction – Finally, execute these operations, also from left to right.

In our problem, the order of operations dictated simplifying the parentheses first, then performing multiplication, and finally, dealing with addition or subtraction. This process ensured that steps were followed correctly, reaching the correct final answer efficiently.

Negative Numbers in Algebra

Working with negative numbers in algebra can be tricky and requires special attention to their rules. Negative numbers affect multiplication and subtraction in particular ways:

• When multiplying two numbers, if one is negative, the result is negative. For our expression, multiplying \(-2\) by \(1\) produces \(-2\).
• Subtraction of a negative number is equivalent to adding the positive of that number. This was crucial in our final step, \(-2 - (-7) = -2 + 7\).

Understanding these properties helps in avoiding errors and ensures accurate calculations throughout the solution process. By managing negatives properly, we can simplify expressions more effectively and reach valid conclusions.