Problem 78 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Evaluate \(3 x+4(y+6)\) when \(x\) is 1 less than the quoticnt of \(y\) and \(3,\) and \(y=15\)

Step-by-Step Solution

Answer

The evaluated result is 96.

Step 1: Understanding the Problem

The sentence 'x is 1 less than the quotient of y and 3' needs to be translated into an equation. The phrase 'quotient of y and 3' means \(y/3\). Then, saying 'x is 1 less' than the quotient refers to subtracting 1 from the quotient. Hence, the translated equation is \(x = y/3 - 1\).

Step 2: Substitute y into the Equation

Now, the given value for y, which is 15, is substituted into the equation from Step 1. This gives: \(x = 15/3 - 1\). After simplifying the right side of the equation, you find that \(x = 5 - 1 = 4\). So, for this given situation, \(x = 4\).

Step 3: Evaluate the Expression

With the values of \(x\) and \(y\) now known, they can be substituted into the given expression: \(3x + 4(y + 6)\). Substituting, this simplifies to \(3(4) + 4(15 + 6)\). Now, it's a matter of simplifying through the order of operations. This gives: \(12 + 4*21 = 12 + 84 = 96\). Hence, the expression evaluates to 96 for the given values of \(x\) and \(y\).

Key Concepts

Writing Equations

Translating verbal sentences into algebraic equations is a fundamental skill in introductory algebra. In this particular problem, we begin with the phrase "x is 1 less than the quotient of y and 3." Such a sentence includes specific keywords that hint at mathematical operations. For instance:

• "Quotient" indicates division.
• "Less than" suggests subtraction.

When translating, identify and represent each part in mathematical terms. The "quotient of y and 3" is written as \( \frac{y}{3} \). "X is 1 less than this quotient" translates to subtracting 1 from \( \frac{y}{3} \), which gives us the expression \( x = \frac{y}{3} - 1 \).
Once you have this equation, it can be used as a basis for solving further parts of the exercise. Writing equations involves recognizing and interpreting these key elements from the word sentence to write a valid algebraic equation.

Substitution Method

The substitution method is a technique used to find unknown variables by replacing them with known values. For this exercise, the given value for \( y \) is 15. Using the equation derived from the previous step, \( x = \frac{y}{3} - 1 \), we substitute 15 into \( y \) yielding:

• \( x = \frac{15}{3} - 1 \)
• Simplifying gives \( x = 5 - 1 \), therefore, \( x = 4 \)

The substitution method makes it easier to solve problems by systematically replacing variables with numbers. Here, knowing \( y \) allowed us to calculate \( x \) directly. This technique is not only instrumental in finding solutions but also in simplifying and evaluating expressions with multiple variables.
In algebra, substitutions can transform complex equations into simpler arithmetic, facilitating a clearer path to the solution.

Order of Operations

Once the variables are substituted, the next step is to evaluate the algebraic expression by following the "order of operations." In mathematics, this order is crucial for obtaining the correct result. The process is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents (powers and square roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our expression \( 3x + 4(y + 6) \), after substitution, it becomes \( 3(4) + 4(15 + 6) \). First, calculate inside the parentheses: \( 15 + 6 = 21 \). Next, do the multiplication: \( 3 \times 4 = 12 \) and \( 4 \times 21 = 84 \). Finally, add these products: \( 12 + 84 = 96 \).
Adhering to the order of operations ensures the calculations withstand mathematical conventions, leading to an accurate evaluation of the expression.