Solving Equations With Multiplication Principle -5/4 * R = -45/8

July 10, 2025 by Scholario Team 65 views

Solving Equations Using the Multiplication Principle: A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Among the various techniques available, the multiplication principle stands out as a powerful tool for isolating variables and finding solutions. This article delves into the application of the multiplication principle, providing a comprehensive guide to solving equations effectively. We will walk through a specific example, breaking down each step to ensure clarity and understanding. Whether you're a student grappling with algebraic concepts or someone seeking to refresh your equation-solving skills, this guide will equip you with the knowledge and confidence to tackle a wide range of mathematical problems.

Understanding the Multiplication Principle

At its core, the multiplication principle states that if you multiply both sides of an equation by the same non-zero number, the equation remains balanced. This principle is rooted in the fundamental concept of equality: whatever operation you perform on one side of an equation, you must perform the same operation on the other side to maintain the balance. This seemingly simple idea is the cornerstone of solving many algebraic equations. By strategically choosing the number to multiply by, we can effectively isolate the variable we are trying to solve for.

To fully grasp the multiplication principle, it’s helpful to think of an equation as a balanced scale. The two sides of the equation represent the two pans of the scale, and the equals sign signifies that the scale is perfectly balanced. If you add or subtract weight from one side, you must do the same on the other side to maintain the balance. Similarly, if you multiply or divide the weight on one side, you must perform the same operation on the other side. The multiplication principle allows us to manipulate equations while preserving their inherent balance, ultimately leading us to the solution.

The multiplication principle is particularly useful when dealing with equations involving fractions or coefficients that are not integers. In such cases, multiplying by the reciprocal of the coefficient of the variable can effectively eliminate the fraction and simplify the equation. The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is 1/2, and the reciprocal of -3/4 is -4/3. When a number is multiplied by its reciprocal, the result is always 1, which allows us to isolate the variable. This technique is crucial for solving equations like the one we will explore in detail below.

Solving the Equation: $- rac{5}{4} r=- rac{45}{8}$

Let's apply the multiplication principle to solve the equation: $- rac{5}{4} r=- rac{45}{8}$ . This equation involves a fractional coefficient for the variable 'r', making the multiplication principle the ideal method for finding the solution. Our goal is to isolate 'r' on one side of the equation, and to do this, we need to eliminate the fraction $- rac{5}{4}$ .

The first step in solving this equation using the multiplication principle is to identify the coefficient of the variable 'r'. In this case, the coefficient is $- rac{5}{4}$ . To isolate 'r', we need to multiply both sides of the equation by the reciprocal of this coefficient. The reciprocal of $- rac{5}{4}$ is $- rac{4}{5}$ . This is because when we multiply a fraction by its reciprocal, the result is 1. Multiplying by the reciprocal will effectively cancel out the coefficient, leaving 'r' by itself on one side of the equation.

Now, we multiply both sides of the equation $- rac{5}{4} r=- rac{45}{8}$ by $- rac{4}{5}$ . This gives us:

$(- rac{4}{5})(- rac{5}{4} r) = (- rac{4}{5})(- rac{45}{8})$

On the left side of the equation, the fractions $- rac{5}{4}$ and $- rac{4}{5}$ cancel each other out, leaving us with just 'r'. This is the direct result of multiplying by the reciprocal. On the right side of the equation, we need to multiply the two fractions. When multiplying fractions, we multiply the numerators together and the denominators together. So, we have:

r = $(- rac{4}{5})(- rac{45}{8})$

Performing the Multiplication and Simplification

To complete the solution, we need to perform the multiplication on the right side of the equation and simplify the resulting fraction. We have:

r = $rac{(-4) imes (-45)}{5 imes 8}$

Multiplying the numerators, we get $(-4) imes (-45) = 180$ . Multiplying the denominators, we get $5 imes 8 = 40$ . So, the equation becomes:

r = $rac{180}{40}$

Now, we need to simplify this fraction. Both the numerator and the denominator are divisible by 10, so we can start by dividing both by 10:

r = $rac{18}{4}$

We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

r = $rac{18 \div 2}{4 \div 2}$

r = $rac{9}{2}$

Therefore, the solution to the equation $- rac{5}{4} r=- rac{45}{8}$ is r = $rac{9}{2}$ . This fraction is in its simplest form, as 9 and 2 have no common factors other than 1. We have successfully isolated 'r' using the multiplication principle and simplified the result to obtain the final answer.

Alternative Simplification Method

In the previous section, we simplified the fraction $rac{180}{40}$ by first dividing by 10 and then by 2. However, there's another approach to simplifying fractions that can sometimes be more efficient. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. In this case, the GCD of 180 and 40 is 20. If we divide both the numerator and the denominator by 20 directly, we can simplify the fraction in a single step.

Starting with the fraction $rac{180}{40}$ , we divide both the numerator and the denominator by 20:

r = $rac{180 \div 20}{40 \div 20}$

r = $rac{9}{2}$

As you can see, this method leads us to the same simplified fraction, $rac{9}{2}$ , but it involves fewer steps. Finding the GCD can sometimes be a bit challenging, but with practice, it becomes a valuable skill for simplifying fractions quickly and efficiently. This alternative method reinforces the idea that there are often multiple paths to solving a mathematical problem, and choosing the most efficient path can save time and effort. The key is to understand the underlying principles and apply them flexibly to the specific problem at hand.

Verification of the Solution

To ensure the accuracy of our solution, it's crucial to verify it by substituting the value we found for 'r' back into the original equation. This step helps us catch any potential errors in our calculations and provides confidence in the correctness of our answer. In this case, we found that r = $rac{9}{2}$ . Let's substitute this value into the original equation:

$- rac{5}{4} r=- rac{45}{8}$

Replacing 'r' with $rac{9}{2}$ , we get:

$- rac{5}{4} ( rac{9}{2})=- rac{45}{8}$

Now, we perform the multiplication on the left side of the equation. When multiplying fractions, we multiply the numerators together and the denominators together:

$- rac{5 imes 9}{4 imes 2}=- rac{45}{8}$

$- rac{45}{8}=- rac{45}{8}$

As we can see, the left side of the equation is equal to the right side of the equation. This confirms that our solution, r = $rac{9}{2}$ , is correct. Verification is an essential step in problem-solving, especially in mathematics. It not only ensures the accuracy of the answer but also reinforces the understanding of the concepts and techniques involved. By verifying our solutions, we develop a deeper understanding of the mathematical principles at play and build confidence in our problem-solving abilities.

Conclusion

In conclusion, the multiplication principle is a powerful and versatile tool for solving equations, particularly those involving fractions. By multiplying both sides of an equation by the reciprocal of the coefficient of the variable, we can effectively isolate the variable and find the solution. In the specific example of $- rac{5}{4} r=- rac{45}{8}$ , we demonstrated the step-by-step application of the multiplication principle, culminating in the solution r = $rac{9}{2}$ . We also explored alternative simplification methods and emphasized the importance of verifying the solution to ensure accuracy.

Mastering the multiplication principle is a crucial step in developing algebraic proficiency. It lays the foundation for solving more complex equations and tackling a wider range of mathematical problems. The ability to manipulate equations while maintaining their balance is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and economics. By understanding and practicing the multiplication principle, you can enhance your problem-solving skills and approach mathematical challenges with confidence.

This article has provided a comprehensive guide to solving equations using the multiplication principle, complete with a detailed example and explanations. By following the steps outlined and practicing regularly, you can master this essential technique and unlock your potential in mathematics. Remember, the key to success in mathematics is consistent practice and a solid understanding of the fundamental principles. With dedication and effort, you can achieve your mathematical goals and excel in your studies.