Problem 82 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Evaluate \(5 x-14 y\) for \(x=3\) and \(y=\frac{1}{2}\) b. Is the number you obtained in part (a) a solution of \(4 w=54-5 w ?\)

Step-by-Step Solution

Answer

a. \(8\) b. No, \(8\) is not a solution for the equation \(4w = 54 - 5w\)

Step 1: Substitute x and y into the Given Equation

Substitute \(x = 3\) and \(y = \frac{1}{2}\) into the expression \(5x - 14y\). So, we have \(5 * 3 - 14 * \frac{1}{2}\) which does simplify to \(15 - 7\).

Step 2: Calculation

Perform the subtraction to find the value of the expression. So, we find \(15 - 7\) equals \(8\).

Step 3: Substitute 8 (Result of part a) into the Second Equation

Substitute \(8\) into the second equation \(4w = 54 - 5w\). We insert \(8\) for \(w\), getting \(4 * 8 = 54 - 5 * 8\). It does simplify to \(32 = 54 - 40\).

Step 4: Validation

Perform the right-hand side operation \(54-40\) to obtain \(14\) and so the left-hand side is not equal the right-hand side (\(32 ≠ 14\)). The assumption is invalid, i.e., \(8\) is not a solution to the equation \(4w = 54 - 5w\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used when solving algebraic expressions and equations. It involves replacing variables with numerical values to simplify the process of evaluation and to find solutions.
In step 1 of the exercise, we substituted the given values for the variables. We had to substitute \(x = 3\) and \(y = \frac{1}{2}\) into the expression \(5x - 14y\).
This gave us \(5 \times 3 - 14 \times \frac{1}{2}\), which simplified to \(15 - 7\). Using substitution makes it easier to handle complex expressions by breaking them down into simple arithmetic operations.

• Start by identifying the values to substitute for each variable.
• Replace the variables in the expression with the given values.
• Simplify the resulting arithmetic expression to find the value.

Equation Solving

Equation solving involves finding the values of variables that make an equation true. This is often the next step after substitution, when we've converted an expression into a simpler form. In the second part of the exercise, we were given the equation \(4w = 54 - 5w\) to solve.

From step 3, the exercise asked us to substitute \(w = 8\) into the given equation. We performed the operations \(4 \times 8\) and \(54 - 5 \times 8\) and simplified both sides to check for equality. Equation solving requires:

• Substitute known values into the equation if applicable.
• Perform operations on both sides of the equation.
• Simplify until you isolate the variable and check for true equality.
• Adjust the variable value if the initial substitution doesn't balance the equation.

Mathematical Validation

Mathematical validation is crucial when determining the correctness of a solution. It involves checking if the calculations and substitutions made in the previous steps result in a true statement.
In step 4, after swapping in \(w = 8\) and performing the calculations, we saw that the left-hand side equaled \(32\) and the right-hand side equaled \(14\).

These values were not equal, indicating that \(w = 8\) was not a valid solution to the equation \(4w = 54 - 5w\). Here are the steps for validation:

• Substitute the found solution back into the original equation.
• Calculate both the left and right sides of the equation independently.
• Compare both sides, looking for equality.
• If the sides are not equal, reassess and adjust your solution.

Validation ensures your solution accurately balances the equation, providing confidence in your work.