Problem 77 - Introductory Algebra for College Students

Question

In Exercises \(73-80,\) evaluate each algebraic expression for the given value of the variable. $$-x^{2}-10 x ; x=-1$$

Step-by-Step Solution

Answer

The answer to the expression \(-x^{2} - 10x\) when \(x = -1\) equals \(9\).

Step 1: Substitution

Substitute the value of the variable \(x\) into the algebraic expression. So \(-x^{2}-10 x\) becomes \(-(-1)^{2} -10*(-1)\.

Step 2: Apply the order of operations (PEMDAS)

Start with the Exponent, then the Multiplication and the Addition. So \(-(-1)^{2} - 10*(-1)\) simplifies to \(-1 - (-10)\).

Step 3: Simplify the expression

Simplify the expression by adding the numbers. \(-1 --10\) simplifies to \(9\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental concept that allows us to replace variables within an expression with specific values. This process helps in evaluating expressions to find their numerical outcomes.

• Understand the expression: In the example, the expression is given as \(-x^{2}-10x\).
• Identify the variable and its value: Here, \(x\) is the variable and its value is \(-1\).
• Replace the variable: Substitute \(x = -1\) into the expression, transforming it into \(-(-1)^{2} - 10(-1)\).

Substitution simplifies the expression by replacing variables, which is a crucial first step in evaluating an algebraic expression. It requires careful attention to ensure that every instance of the variable is replaced correctly.

Order of Operations

The order of operations is the rule set that dictates the correct sequence to evaluate parts of a mathematical expression. The acronym PEMDAS is often used to remember this order:

• P: Parentheses - perform operations within parentheses first.
• E: Exponents - solve exponential expressions.
• MD: Multiplication and Division - perform from left to right.
• AS: Addition and Subtraction - perform from left to right.

In the provided example, the order of operations is crucial for solving the expression: - Start by calculating the exponent, \((-1)^{2}\), which results in \(1\).- Next, perform the multiplications: \(-1 \times 1\) and \(-10 \times -1\), resulting in \(-1\) and \(10\), respectively.- Finally, perform the subtraction \(-1 - (-10)\), which simplifies to \(-1 + 10\), giving the result.Remembering PEMDAS ensures that each operation is executed in the correct order, avoiding common mistakes.

Evaluating Expressions

Evaluating expressions involves simplifying them to arrive at a single numerical value. This process merges the steps of substitution and applying the order of operations.- Begin by substituting the variable with the given value, as explained in the substitution section.- Apply the order of operations to transform the expression into a simplified form. This includes calculating exponents, then performing multiplications, and finally addressing additions or subtractions.- The final step is to simplify all operations into one number.In our example, we substitute \(-1\) for \(x\), use the order of operations, and calculate:- Start with the exponent: \((-1)^2 = 1\).- Multiply: \(-1 \times 1 = -1\) and \(-10 \times -1 = 10\).- Complete the expression: \(-1 + 10 = 9\).Evaluating expressions is essentially piecing together everything you know about substitution and order of operations to find the answer. The result here is easiest: the expression evaluates to \(9\).