Problem 24 - Elementary and Intermediate Algebra

Question

Evaluate using a calculator. $$ 19 x y-9 x+13 y, \text { for } x=87 \text { and } y=29 $$

Step-by-Step Solution

Answer

47525

Step 1 - Substitute the Values of x and y

Substitute the given values of x and y into the expression. The expression is: $$ 19xy - 9x + 13y $$ Given: x = 87 and y = 29.

Step 2 - Compute the Value of Each Term Separately

First, compute the value of the term 19xy. Multiply 19, x, and y together: $$ 19 \times 87 \times 29 $$ Second, compute the value of the term 9x. Multiply 9 by x: $$ 9 \times 87 $$ Finally, compute the value of the term 13y. Multiply 13 by y: $$ 13 \times 29 $$

Step 3 - Calculate the Products

Compute the products: $$ 19 \times 87 \times 29 = 47931 $$ $$ 9 \times 87 = 783 $$ $$ 13 \times 29 = 377 $$

Step 4 - Substitute the Products Back into the Expression

Substitute the computed values back into the original expression: $$ 47931 - 783 + 377 $$

Step 5 - Perform the Addition and Subtraction

Subtract 783 from 47931 and then add 377 to the result: $$ 47931 - 783 + 377 = 47148 + 377 = 47525 $$

Key Concepts

Substitution Method

The substitution method is used to replace variables in an algebraic expression with their given numerical values. This makes the expression easier to evaluate.
In this exercise, we substitute the values of \(x = 87\) and \(y = 29\) into the expression \(19xy - 9x + 13y\).
Here’s a simple way to handle it:

• Identify the variables (\(x\) and \(y\)) in the expression.
• Replace \(x\) and \(y\) with their given values.

This step transforms an abstract variable expression into a concrete numerical one.

Multiplication

Multiplication is the mathematical process of combining equal groups. In this exercise, it is a key operation to evaluate each term:

First, consider the term \(19xy\). Multiply the constants and variables together:
\( 19 \times 87 \times 29 \rightarrow 19 \times 87 = 1653 \rightarrow 1653 \times 29 = 47931 \)

Next, consider the term \(9x\). Multiply 9 by 87:
\( 9 \times 87 = 783 \)

Finally, consider the term \(13y\). Multiply 13 by 29:
\( 13 \times 29 = 377 \)

By carefully performing each multiplication step by step, you ensure accuracy in your calculations.

Order of Operations

Order of operations is crucial for ensuring consistency in mathematical evaluations. The sequence in which operations (addition, subtraction, multiplication, division) are performed impacts the final result. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For this exercise:

After substituting and multiplying the individual terms, we have:\(47931 - 783 + 377 \). Now follow the order:

• First, perform the subtraction: \(47931 - 783 = 47148 \)


• Then, perform the addition: \(47148 + 377 = 47525 \)

This methodical approach ensures that you get the correct final result, \(47525\).