Problem 19 - Introductory Algebra for College Students

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 into the expression

Replace \(x\) with 7 and \(y\) with 5 in the expression \(4x - 3y\). The equation becomes \(4(7) - 3(5)\).

Step 2: Perform Multiplication

Use the order of operations (PEMDAS/BODMAS) to perform multiplication before subtraction. Multiply 4 and 7 to get 28, and 3 and 5 to get 15. The equation now becomes \(28 - 15\).

Step 3: Perform Subtraction

Finally, subtract 15 from 28 to get the result of the expression. 28 - 15 equals 13.

Key Concepts

Substitution in Algebraic Expressions

Substitution is a cornerstone of algebra, allowing us to evaluate expressions by replacing variables with given numerical values. This process simplifies the expression to a numerical equation that can be solved using basic arithmetic. In our example, the algebraic expression is \(4x - 3y\). Here, \(x\) is substituted with 7 and \(y\) is replaced with 5, transforming the expression into \(4(7) - 3(5)\).

• Identify variables in your expression. In this example, they are \(x\) and \(y\).
• Replace each variable with the corresponding given number.
• Rewrite the expression with the substituted values.

This step ensures that you're set up for performing calculations with exact numbers rather than symbols.

Understanding Order of Operations

Order of operations is a critical concept in mathematics that dictates the sequence in which operations are performed in an expression. This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)). Mathematicians may also use BODMAS or BIDMAS. In the expression \(4(7) - 3(5)\), we first carry out the multiplications:

• Multiply: Perform multiplication before any addition or subtraction. Here, calculate \(4 \times 7 = 28\) and \(3 \times 5 = 15\).
• Subtract: After multiplication, proceed to subtraction. Thus, we now solve \(28 - 15\).

Following this order is crucial to avoid errors and reach the correct solution.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are foundational concepts in mathematics and are used repeatedly when working with algebraic expressions. In our example, we focus on multiplication and subtraction.- **Multiplication**: Calculating the product of numbers, \(4 \times 7 = 28\) and \(3 \times 5 = 15\), is the first step after substitution. Ensuring accuracy in multiplication is vital, as it sets the stage for the next operations.- **Subtraction**: Find the difference between numbers, which is the final step of our example. Subtract \(15\) from \(28\) to get the result, \(13\).Each of these operations must be performed after careful consideration of substitution and the order of operations to solve expressions accurately.