Question

Explain how to evaluate an expression. $$3 y-5(y+2) ; y=4$$

Short answer

Answer: The result is $$-18$$.

Step-by-step solution

Identify the expression and the given value of the variable

The given expression is $$3y - 5(y + 2)$$ and the value of the variable $$y$$ is given as $$y = 4$$.

Substitute the value of the variable into the expression

Replace $$y$$ in the expression with the given value, which is $$4$$: $$3(4) - 5(4 + 2)$$

Perform operations inside parentheses

Calculate the expressions inside the parentheses: $$3(4) - 5(6)$$

Apply the multiplication

Multiply the numbers outside the parentheses by the numbers inside the parentheses: $$12 - 5(6)$$ $$12 - 30$$

Perform subtraction

Subtract the second term from the first term: $$12 - 30 = -18$$ The evaluated expression is $$-18$$.

Key concepts

Substitution Method

The substitution method is a crucial part of evaluating algebraic expressions. It involves replacing the variable in an expression with its given numerical value. For example, in the expression \(3y - 5(y + 2)\), if we know that \(y = 4\), we substitute \(4\) for every instance of \(y\).

Here's how the substitution step works:

• Locate every variable in the expression.
• Replace the variable with the given number.
• Ensure that the substitution is clear, such as by using parentheses: \(3(4) - 5((4) + 2)\).

Substitution is the first step and sets the stage for further operations to solve the expression.

Order of Operations

The order of operations is a rule that tells us the sequence in which to solve different parts of a mathematical expression. It is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

To apply these rules to an expression like \(3(4) - 5(4 + 2)\), you follow these steps:

• Start with calculations inside the parentheses: \(4 + 2\).
• Proceed to multiplication or division: \(3(4)\) and \(5(6)\).
• Finish with addition or subtraction, which in this case is just a final subtraction: \(12 - 30\).

Failing to follow the order of operations can lead to incorrect results, so it's important to apply them meticulously.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. The expression \(3y - 5(y + 2)\) is an example of an algebraic expression that includes the variable \(y\), constants (3 and 5), operations (multiplication and subtraction), and parentheses.

Understanding algebraic expressions involves:

• Identifying terms, which are parts of the expression separated by addition or subtraction.
• Recognizing coefficients, which are the numbers multiplied by the variables (like \(3\) in \(3y\)).
• Understanding the role of parentheses in grouping terms.

The ability to work with algebraic expressions is fundamental to solving problems in algebra and beyond.