Problem 127 - Prealgebra and Introductory Algebra

Question

Is \(-\frac{4}{5}\) a solution of the equation \(\frac{5}{4} n=-1 ?\)

Step-by-Step Solution

Answer

Yes, \(-\frac{4}{5}\) is a solution to the equation \(\frac{5}{4} n=-1\).

Step 1 : Substitute the value

Substitute \( n = - \frac{4}{5} \) into the equation \( \frac{5}{4} n = -1 \) .

Step 2 : Simplify the equation

When you substitute, the equation becomes \( \frac{5}{4} (-\frac{4}{5} ) = -1 \) that simplifies to \(-1 = -1\).

Step 3: Confirm the solution

Both sides of the equation equal -1, thus confirming that \( n = - \frac{4}{5} \) is indeed a solution of the equation \( \frac{5}{4} n = -1 \).

Key Concepts

Substitution Method

The substitution method is a foundational technique used for solving algebraic equations, including linear equations. It involves replacing a variable in an equation with a given number or expression. When a problem asks whether a certain value is a solution to an equation, you perform a substitution to determine its validity.

In the exercise, the question was to verify if \( -\frac{4}{5} \) is a solution to the equation \( \frac{5}{4} n = -1 \). To use the substitution method here, you simply replace the variable \( n \) with \( -\frac{4}{5} \) and calculate the result.

It's crucial to follow the proper order of operations when performing the substitution. First replace the variable, then perform any multiplications or divisions as required. This approach simplifies the equation and allows us to see if we arrive at a true statement, verifying the proposed solution.

Simplifying Fractions

Simplifying fractions is a crucial step in solving equations, as it can transform complex expressions into simpler forms that are easier to understand and manipulate. This process often involves reducing the fraction to its lowest terms by dividing the numerator and the denominator by their greatest common factor.

However, when working with equations, simplifying fractions also refers to the process of performing arithmetic operations correctly. For example, when a fraction multiplies another fraction, you multiply the numerators together and the denominators together.

In the given example, simplifying \( \frac{5}{4} \times -\frac{4}{5} \) involves multiplying the numerators (5 x -4) and the denominators (4 x 5), which results in \( -\frac{20}{20} \). This simplifies to -1, since -20 divided by 20 equals -1. Recognizing these simplifications can make solving equations quicker and less prone to error.

Verifying Solutions

Verifying solutions is the final, but one of the most important steps in the problem solving process. This involves checking that the proposed solution indeed satisfies the original equation. After applying previous techniques, such as substitution and simplification, you must look at both sides of the equation to confirm they have equal values.

Using our example, after substituting and simplifying, we determined that both sides of the equation equal -1. This demonstrates the proposed value of \( n = -\frac{4}{5} \) satisfies the equation \( \frac{5}{4} n = -1 \), making it a valid solution.

Verification ensures that no mistakes were made during the process and that the solution is mathematically sound. It also reinforces understanding of the relationship between the variables and constants within equations, bolstering one's confidence in working with algebraic expressions.