Problem 77 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Evaluate \(4 x+3(y+5)\) when \(x\) is 1 less than the quotient of \(y\) and 4 and \(y=12\)

Step-by-Step Solution

Answer

The evaluated result of the expression is 59.

Step 1: Translate the information into mathematical form

The first instruction is asking to write the sentence as an equation. The sentence 'x is 1 less than the quotient of \(y\) and 4' can be written as \(x = \frac{y}{4} - 1)\. Also, we know that \(y=12\).

Step 2: Substitution of variables

Now we need to substitute the value of \(y\) we know into the expression for \(x\). So we have \(x = \frac{12}{4} - 1\). This simplifies to \(x = 2\), so now we know both \(x = 2\) and \(y = 12\).

Step 3: Evaluating the given expression

Now we need to substitute the values we've found for \(x\) and \(y\) into the original expression. We get \(4 * 2 + 3 * (12 + 5)\).

Step 4: Performing the operations

The next step is to perform the operations in the expression according to the order of operations (also known as BODMAS or PEMDAS). Prioritizing multiplication and addition as per the order, the equation simplifies to \(8 + 3 * 17\), which then becomes \(8 + 51\).

Step 5: Final Simplification

Finally, adding our results together yields \(8 + 51 = 59\).

Key Concepts

Equation Writing

When translating a sentence into a mathematical equation, it's important to understand the relationships described in words. Consider the sentence: "x is 1 less than the quotient of \(y\) and 4." Here, we need to identify key components: what "quotient" means and how to express "1 less than."

The quotient involves division, so when saying "the quotient of \(y\) and 4," we mean dividing \(y\) by 4. "1 less than" indicates subtraction by one unit. Thus, this relationship can be expressed as an algebraic equation: \(x = \frac{y}{4} - 1\). This equation succinctly represents the relationship given in the problem, ready for further evaluation.

Translating words into algebraic expressions involves:

• Identifying key mathematical operations (e.g., 'less than' indicates subtraction).
• Formulating expressions that represent those operations mathematically.

With this framework in place, you're equipped to solve the equation using known values.

Substitution

Substitution is a valuable technique in algebra that allows you to simplify expressions by replacing variables with known values. By substituting, you can solve equations and find specific numerical solutions.

In our exercise, the variable \(y\) is known to be 12. Having rewritten our initial sentence into an algebraic expression, we have \(x = \frac{y}{4} - 1\). We can substitute the value of \(y = 12\) into this expression to solve for \(x\). This means:

• Replacing \(y\) with 12, the equation becomes \(x = \frac{12}{4} - 1\).
• The division is performed first, resulting in \(x = 3 - 1\).
• After simplifying, we find \(x = 2\).

Substitution helps streamline the process of solving equations by using known quantities.

To effectively apply substitution:

• Ensure the equation is set up correctly before substituting.
• Verify each step of your calculations to ensure accuracy.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), ensures that mathematical expressions are consistently evaluated. Failing to follow this order can lead to incorrect results.

For instance, in the expression \(4x + 3(y + 5)\) given our \(x = 2\) and \(y = 12\), it's essential to:

• First, perform the operation inside the parentheses: \(y + 5 = 12 + 5 = 17\).
• Then, conduct the multiplication operations: \(4 \times 2 = 8\) and \(3 \times 17 = 51\).
• Finally, carry out the addition: \(8 + 51 = 59\).

Each step follows a specific sequence dictated by the order of operations, allowing for accurate simplification and evaluation of expressions.

Remember: Adhering to the correct sequence is crucial for solving expressions correctly. Always verify each intermediary result to ensure precision.