Problem 82 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. 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) The value for equation \(5 x-14 y\) with \(x=3\) & \(y=\frac{1}{2}\) is 8. b) The number obtained in part (a) which is 8, is not a solution to the equation \(4 w=54-5 w\).

Step 1: Substitute Values in Equation

First, substitute the given values into the first equation, which means replacing \(x\) value with 3 and \(y\) value with \(\frac{1}{2}\). So, \(5x - 14y = 5(3) - 14(\frac{1}{2})\).

Step 2: Simplify the Equation

Perform the multiplication and subtraction to simplify the equation. \(5x - 14y = 15 - 7\). The result is 8.

Step 3: Test the Solution in another Equation

Substitute \(w\) with the value obtained in the previous step (8) into the second equation, \(4w = 54 - 5w\). If both sides of the equation are equal, then the number obtained in part (a) is a solution to the equation.

Step 4: Simplify the Equation

After substitution we have, \(4(8) = 54 - 5(8)\). So, \(32 = 54 -40\). The result is \( 32 = 14\). Both sides of the equation are not equal. Hence, the value obtained in part (a) is not a solution to this equation.

Key Concepts

Substitution Method

The substitution method is a foundational technique in algebra used for solving systems of equations, where one variable is replaced with an equivalent expression containing the other variable. But its usefulness extends beyond systems of equations. For instance, in the given exercise, substitution is used to evaluate an expression by replacing variables with their given values.

Let's apply the substitution method to the provided problem: The initial instruction is to evaluate the expression \(5x - 14y\) for \(x=3\) and \(y=\frac{1}{2}\). To do this, we substitute x with 3 and y with \(\frac{1}{2}\) directly into the expression. Essentially, wherever we see \(x\) in the expression, we replace it with 3, and wherever we see \(y\), we replace it with \(\frac{1}{2}\).

Breaking Down Substitution

In practice, this looks like \(5(3) - 14(\frac{1}{2})\), which is directly applying the given values to our expression. It's crucial in algebra to perform substitution accurately to avoid errors in calculation and to ensure that the simplification process that follows can be done correctly.

Simplifying Equations

Once we have substituted our variables, we simplify the equations. Simplifying is, in essence, performing all the manageable operations to bring the equation to its simplest form, including addition, subtraction, multiplication, and division. In the provided problem, after substitution, we are left with \(5(3) - 14(\frac{1}{2})\).

The next step is to perform the multiplication: \(5 \times 3 = 15\) and \(14 \times \frac{1}{2} = 7\). Now, we combine these products, subtracting 7 from 15, which yields 8. This simplified result represents the evaluation of the initial expression with the given values for \(x\) and \(y\).

Why Simplify?

The purpose of simplifying equations is to make them more accessible and more straightforward to comprehend or to prepare them for further calculations or applications, such as testing the result in another equation, which is precisely what we'll need to do in the following concept of verifying solutions.

Verifying Solutions

Verifying solutions is a critical step in the process. It involves taking the solutions we've found and ensuring they satisfy the original equations or the conditions set forth by the problem. It's essentially a 'check' to guarantee accuracy.

In context with the example at hand, the solution obtained from part (a), which was 8, is now tested in another equation \(4w = 54 - 5w\) to see if it is, in fact, a correct solution. We substitute \(w\) with 8, yielding the equation \(4(8) = 54 - 5(8)\).

By simplifying both sides of this new equation, we find that they do not equate (\(32 eq 14\)), demonstrating that the number we attained from part (a) is not a solution to this second equation.

The Value of Verification

Verifying not only serves as proof that we have done the right calculations but also prevents the perpetuation of errors in subsequent tasks or analyses. It is the last step to ensure that our answer from one part of the problem is applicable to another, ensuring consistency and reliability in our mathematical reasoning.