Problem 107 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$\frac{5 m-1}{6}=\frac{3 m-2}{4},-4$$

Step-by-Step Solution

Answer

Yes, -4 is a solution to the equation.

Step 1: Substitute Value

Substitute \(m = -4\) into the equation \(\frac{5 m-1}{6}=\frac{3 m-2}{4}\). This will give: \(\frac{5(-4)-1}{6}=\frac{3(-4)-2}{4}\)

Step 2: Simplify Both Sides

On simplifying both sides, you will have: \(\frac{-21}{6} = \frac{-14}{4}\). In other terms this is \(-\frac{7}{2} = -\frac{7}{2}\).

Step 3: Verify Equation

Observing the final result, both sides of the equation are equal, confirming -4 is indeed a solution of the equation.

Key Concepts

Equation Solving

Equation solving is the process of finding the unknown value that makes an equation true. An equation is a mathematical statement showing that two expressions are equal, indicated by the symbol \( = \). In algebra, the goal is often to find the value of the variable that satisfies the equation.

To solve equations effectively, you need to understand the type of equation and the operations involved. This might involve basic arithmetic or more complex operations like fractions and exponents. Here's a simple approach to follow:

• Identify the equation's components and structure. Check for operations like addition, multiplication, and fractions.
• Isolate the variable on one side. This might involve manipulating one or both sides of the equation.
• Use inverse operations to simplify the equation, steadily working toward isolating the variable.
• Check your solution by substituting the value back into the original equation to verify accuracy.

Equation solving can sometimes involve more than one method or multiple steps. Paying close attention to details is crucial in reaching the correct solution.

Substitution Method

The substitution method is widely used in equation solving, especially useful when dealing with linear equations. It's a technique where you replace one variable with a given value to determine the other variables or to check if a number is a solution.

In the original exercise, we used substitution by plugging the given value, \(m = -4\), into the equation \(\frac{5m - 1}{6} = \frac{3m - 2}{4}\). We replaced \(m\) with \(-4\) and performed the necessary arithmetic to evaluate both sides:

• Left Side: \(\frac{5(-4) - 1}{6} = \frac{-21}{6}\)
• Right Side: \(\frac{3(-4) - 2}{4} = \frac{-14}{4}\)

Once substitution is complete, it’s important to simplify both expressions. If both sides of the equation are equal after simplification, the given value is a solution. This method is valuable as it is systematic and straightforward, helping verify potential solutions efficiently.

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, often required when solving equations involving fractional expressions. It involves reducing a fraction to its simplest form where the numerator and denominator have no common factors other than 1.

To simplify a fraction, follow these steps:

• Determine the greatest common factor (GCF) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCF.
• Express the resulting fraction, confirming there are no further common factors.

In our example, after substituting \(m = -4\), we obtained \(\frac{-21}{6}\) and \(\frac{-14}{4}\). By simplifying, both fractions reduce to \(-\frac{7}{2}\). This highlights that simplifying fractions not only makes calculations easier but also confirms equality or solutions in equations.

Remember, practicing fraction simplification strengthens overall mathematical proficiency, aiding in various mathematical tasks and problem-solving scenarios.