Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression when \(x=3, y=-2\), and \(z=4$$x^{2}-5 y+4 z\)

Step-by-Step Solution

Answer

The value of the expression when \(x=3\), \(y=-2\), and \(z=4\) is 35.

Step 1: Substitute the Values

Firstly, substitute the values of variables into the expression, this means replacing \(x\) with 3, \(y\) with -2, and \(z\) with 4 in the expression which results in the new expression: \((3)^{2}-5(-2)+4(4)\)

Step 2: Simplify Exponents and Multiplications

Next, simplify the exponent, which means calculating \(3^2\) and then continue by performing the multiplication for the other two terms. This results in: \(9+10+16\)

Step 3: Perform the Addition

Lastly, simplify the expression by performing the addition of all terms. This results in: \(9+10+16 = 35\)

Key Concepts

Substitution Method

The substitution method is a foundational technique used in algebra for evaluating expressions. It involves replacing the variables in an algebraic expression with their corresponding numerical values. In the given exercise, you are asked to evaluate the expression when certain values for the variables are provided:

For example, with given values of 3 for x, -2 for y, and 4 for z, you replace each variable in the expression with these numbers. It's essential to be careful and retain the structure of the original expression, including any exponents or coefficients attached to the variables. The success of this method relies on accurate replacement and careful attention to any signs (positive or negative) associated with the numbers you're substituting.

This step is crucial as it sets the stage for the subsequent arithmetic operations that will lead to the final value of the expression.

Simplifying Exponents

After substituting the variable values, the next step in evaluating the algebraic expression is simplifying exponents. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^2\), 3 is the base, and 2 is the exponent, which tells us that 3 is to be multiplied by itself once. Simplifying \(3^2\) results in \(9\), as \(3 \times 3 = 9\).

In this exercise, only the variable x had an exponent. No other variable had an exponent, which made the process straightforward. Ensuring that each exponent is simplified before moving on to other operations helps in keeping the calculations organized and reduces the possibility of making errors. Once the exponents are simplified, you are left with basic numbers, which you'll then use in further arithmetic operations to solve the expression.

Arithmetic Operations

With exponents simplified, the expression now consists of just numbers and basic arithmetic operations. At this stage, you follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Since there are no parentheses or division in our sample expression, you start with multiplications as seen in \(4 \times 4\), which equals \(16\). After multiplication, move on to addition and subtraction.

For the given exercise, after you've simplified the exponent and carried out the multiplication, you will add the terms together as follows: \(9 + 10 + 16\). Adding these numbers in sequence, you arrive at the sum of 35, which completes the evaluation of the expression.

Understanding and correctly applying arithmetic operations are crucial for solving algebraic expressions accurately. Always consider the signs of numbers and maintain the order of operations for the best results.