Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate \(3 x+4(y+6)\) when \(x\) is 1 less than the quotient of \(y\) and \(3,\) and \(y=15\)

Step-by-Step Solution

Answer

The evaluation of the expression \(3x + 4(y + 6)\) when \(x\) is 1 less than the quotient of \(y\) and 3, and \(y=15\) is 96.

Step 1: Evaluating the value of x

The problem states x is 1 less than the quotient of y and 3, which can be written as \(x = \frac{y}{3} - 1\). Substituting \(y = 15\) into this equation: \(x = \frac{15}{3} - 1 = 5 - 1 = 4\).

Step 2: Substituting x and y into the expression

Now we substitute \(x = 4\) and \(y = 15\) into the expression \(3x + 4(y + 6)\). Therefore the expression becomes: \(3*4 + 4*(15 + 6) = 12 + 4*21 = 12 + 84 = 96\).

Step 3: Final Evaluation

So, after the substitution and evaluation of the expression with the given conditions, the final result is 96.

Key Concepts

Quotients

Quotients are an essential part of arithmetic and algebra. Think of a quotient as the result of a division problem. For instance, if you have a division operation like 15 divided by 3, the quotient is 5. In algebraic contexts, quotients often involve variables.
In the given exercise, the quotient is used to find the value of \(x\). The problem states that \(x\) is 1 less than the quotient of \(y\) divided by 3. This means you first find the quotient by dividing \(y\) by 3, and then subtract 1 from that result.

Using quotients helps solve complex expressions in a structured way. It breaks down larger problems into smaller, more manageable parts. By understanding and calculating quotients correctly, you'll be better equipped to substitute and evaluate these results in larger algebraic expressions.

Variable Substitution

Variable substitution is a key technique in algebra, where you replace variables with given numerical values or expressions. This makes it possible to evaluate algebraic expressions with specific values in place.
In our exercise, we start by determining the variable \(x\) based on the given condition. Here, \(x\) is said to be 1 less than the quotient of \(y\) and 3. Once we calculated the quotient to be 5, we subtract 1, giving us \(x = 4\).

Next, we substitute this value for \(x\), along with the given \(y = 15\), into the algebraic expression \(3x + 4(y + 6)\).

• First, replace \(x\) with 4, transforming the expression into \(3(4) + 4(y + 6)\).
• Then, replace \(y\) with 15 so that you have \(3(4) + 4(15 + 6)\).

Variable substitution simplifies expressions, allowing you to calculate specific numerical results from variable-based equations.

Numerical Evaluation

After substituting the known values of the variables into the expression, the next step is numerical evaluation. This involves simplifying the expression by performing arithmetic operations like addition, subtraction, multiplication, and division.
For the expression \(3x + 4(y + 6)\) after substituting \(x = 4\) and \(y = 15\):

• First, calculate \(3 \times 4\) to get 12.
• Then solve the inside of the parentheses: \((15 + 6)\), which equals 21.
• Next, calculate \(4 \times 21\) to get 84.

Now, add these results together: \(12 + 84\) to obtain the final result of 96.

Numerical evaluation is crucial because it gives you the final numerical answer after all substitutions and calculations. It's the last step where you see the concrete outcome of working through an algebraic problem.