Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$4 x-3 y$$

Step-by-Step Solution

Answer

The result of the expression \(4x - 3y\) for \(x=7\) and \(y=5\) is 13.

Step 1: Substitute the values of x and y into the expression

Replace x with 7 and y with 5 in the original expression. This will give the new expression \(4*7 - 3*5\).

Step 2: Perform Multiplication First

According to the order of operations, perform the multiplication first. So the expression \(4*7 - 3*5\) becomes \(28 - 15\).

Step 3: Subtract

Lastly, perform the subtraction operation: \(28 - 15 = 13\). So, the result of the expression \(4x - 3y\) for \(x=7\) and \(y=5\) is 13.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It determines the correct sequence for performing different arithmetic operations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), these rules ensure consistency in calculations. In this exercise, applying the order of operations means we handle multiplication before subtraction. Thus when evaluating the expression \(4x - 3y\), after substituting the given values, we first calculate \(4 \times 7\) and \(3 \times 5\) before proceeding to subtraction. By following the order of operations, we avoid common errors and ensure the expression is solved correctly.

Substitution

Substitution is an essential technique in algebra that involves replacing variables with their corresponding numerical values. This is particularly useful when given specific values for variables in an expression. In this example, we substitute \(x\) with 7 and \(y\) with 5 in the expression \(4x - 3y\).

• Start by identifying the variables and their given values.
• Replace each instance of the variable in the expression with its numerical value.

So, the original expression \(4x - 3y\) becomes \(4 \times 7 - 3 \times 5\). Substitution simplifies the expression by converting it from variable terms into pure numbers, making it easier to perform further calculations.

Multiplication and Subtraction

Once substitution has transformed the expression into numbers, multiplication and subtraction are performed. According to the order of operations, multiplication comes before subtraction. Therefore, in the expression \(4 \times 7 - 3 \times 5\), calculate each multiplication part first:

• Multiply 4 by 7 to get 28.
• Multiply 3 by 5 to get 15.

After obtaining these products, proceed with subtraction:

• Subtract 15 from 28 to arrive at the final solution of 13.

This clear-cut process helps in avoiding errors by systematically working through each arithmetic operation in turn.