Problem 78 - Introductory Algebra for College Students

Question

In Exercises \(73-80,\) evaluate each algebraic expression for the given value of the variable. $$-x^{2}-14 x ; x=-1$$

Step-by-Step Solution

Answer

The value of the given algebraic expression \(-x^{2}-14 x\) for \(x = -1\) is 13.

Step 1: Identify the Algebraic Expression and the Value of the Variable

The algebraic expression given in the problem is \(-x^{2}-14 x\) and the value of the variable \(x\) given is \(-1\).

Step 2: Substitute the Value of the Variable in the Algebraic Expression

Replace \(x\) with \(-1\) in \(-x^{2}-14 x\) to get \(-(-1)^{2} - 14*(-1)\).

Step 3: Carrying out the Operation

First apply the power operation to get \(-1^2 = -1\), then perform the multiplication operation to get \(14*1 = 14\). This results in the expression \(-1+14\).

Step 4: Simplifying the expression

Finally, perform the addition operation on the resulting expression to get the final result. So, \(-1 + 14 = 13\).

Key Concepts

Algebraic Expression

An algebraic expression is like a mathematical phrase that can include numbers, variables, and operation symbols. Variables in an algebraic expression are often represented by letters, such as the letter 'x' which can take on various values. In our exercise, the algebraic expression given is \( -x^{2}-14x \). To understand these expressions, think of them as a recipe for baking, where the ingredients are numbers and variables, and the cooking instructions are the operations (like addition and subtraction). Just as following a recipe gives you a delicious cake, correctly evaluating an algebraic expression gives you a numerical answer.

Substitution Method

The substitution method is a way to find out what an algebraic expression is worth by replacing its variables with specific values. If an algebraic expression is a mask at a masquerade, the value you substitute for the variable is the person behind that mask. By substituting, we reveal the true identity of the expression. In the exercise, we are asked to evaluate \( -x^{2}-14x \) for \( x=-1 \). Here, we follow the steps of the substitution. We take \( x \) out and put \( -1 \) in its place, much like changing the batteries in a flashlight to make it work. The process sheds light on the real value of the expression when \( x \) becomes \( -1 \).

Simplifying Expressions

Simplifying expressions is similar to tidying up a messy room; it's about making the expression cleaner and easier to understand. It involves performing the operations in the correct order — remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) to guide you. This process will reveal what the expression simplifies to with the substituted values. In our exercise, after substituting \( -1 \) for \( x \) we perform the operations and simplify to get \( -1+14 \). The expression no longer has a variable and is easier to work with, like having a clean room where you know exactly where everything is. Once the expression is simplified, it shows that the final answer is 13, which is as satisfying as looking at a well-organized space after a good clean-up.