Solving For X In The Equation 2/3 = (x-1)/5
In the realm of mathematics, solving for variables is a fundamental skill. This article delves into the process of solving for x in the equation 2/3 = (x-1)/5. We will explore the underlying principles, step-by-step solutions, and practical applications of this type of equation. Whether you are a student learning algebra or simply seeking to refresh your mathematical skills, this comprehensive guide will provide you with the knowledge and confidence to tackle similar problems.
Understanding the Basics of Algebraic Equations
Before diving into the specifics of solving 2/3 = (x-1)/5, it is crucial to grasp the fundamental concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true.
In our case, the equation 2/3 = (x-1)/5 involves a single variable, x. The left-hand side (LHS) of the equation is the fraction 2/3, while the right-hand side (RHS) is the expression (x-1)/5. Our objective is to isolate x on one side of the equation, thereby determining its value. To achieve this, we will employ various algebraic techniques, such as cross-multiplication and the application of inverse operations.
Equations like this are known as linear equations, which are equations where the highest power of the variable is 1. Linear equations are ubiquitous in mathematics and have numerous applications in various fields, including physics, engineering, and economics. Mastering the art of solving linear equations is, therefore, an essential skill for anyone pursuing studies or careers in these areas. Understanding the properties of equality is critical. The properties of equality state that you can perform the same operation on both sides of an equation without changing its solution. These properties include the addition property of equality, the subtraction property of equality, the multiplication property of equality, and the division property of equality. We will utilize these properties throughout the solution process to maintain the balance of the equation while isolating x. The concept of inverse operations is also fundamental. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. When solving equations, we use inverse operations to isolate the variable. For instance, if a number is added to x, we subtract that number from both sides of the equation to isolate x. If x is multiplied by a number, we divide both sides of the equation by that number to isolate x. By understanding these foundational concepts, we lay the groundwork for successfully solving the equation 2/3 = (x-1)/5 and similar algebraic problems.
Step-by-Step Solution to 2/3 = (x-1)/5
Now, let's delve into the step-by-step solution of the equation 2/3 = (x-1)/5. We will meticulously break down each step, explaining the reasoning behind it to ensure clarity and comprehension.
Step 1: Cross-Multiplication
The first step in solving this equation is to eliminate the fractions. We achieve this by employing the technique of cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In our case, we multiply 2 (the numerator of the left-hand side) by 5 (the denominator of the right-hand side) and 3 (the denominator of the left-hand side) by (x-1) (the numerator of the right-hand side).
This gives us the equation:
2 * 5 = 3 * (x - 1)
Simplifying this, we get:
10 = 3(x - 1)
Cross-multiplication is a powerful technique that simplifies equations involving fractions. It is based on the property that if a/b = c/d, then ad = bc. This property holds true because multiplying both sides of the equation a/b = c/d by bd results in ad = bc. By applying cross-multiplication, we transform the original equation into a more manageable form, free of fractions, which makes it easier to proceed with solving for x. This step is crucial as it sets the stage for the subsequent steps, where we will distribute, combine like terms, and isolate x. Without cross-multiplication, the presence of fractions would make the equation significantly more complex to solve. Therefore, mastering this technique is essential for tackling a wide range of algebraic equations involving fractions.
Step 2: Distribute on the Right-Hand Side
Next, we need to eliminate the parentheses on the right-hand side of the equation. This involves applying the distributive property, which states that a(b + c) = ab + ac. In our case, we need to distribute the 3 across the (x - 1) term.
So, 3(x - 1) becomes 3 * x - 3 * 1, which simplifies to 3x - 3. Now our equation looks like this:
10 = 3x - 3
The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. It allows us to multiply a single term by a group of terms enclosed in parentheses. In this particular step, distributing the 3 is crucial because it allows us to separate the x term from the constant term within the parentheses. This separation is necessary for us to eventually isolate x on one side of the equation. Without distributing, we would be unable to combine the constant terms and effectively solve for x. The distributive property is not only applicable to this type of equation but is also a cornerstone of more advanced algebraic manipulations. It is used in expanding polynomials, factoring expressions, and simplifying complex equations. Therefore, a solid understanding of the distributive property is essential for success in algebra and beyond.
Step 3: Isolate the Term with x
Our next goal is to isolate the term containing x (which is 3x) on one side of the equation. To do this, we need to eliminate the -3 on the right-hand side. We can achieve this by adding 3 to both sides of the equation. This is based on the addition property of equality, which states that adding the same value to both sides of an equation preserves the equality.
Adding 3 to both sides, we get:
10 + 3 = 3x - 3 + 3
Simplifying this, we have:
13 = 3x
Isolating the term with x is a crucial step in solving for x because it brings us closer to having x by itself on one side of the equation. By adding 3 to both sides, we effectively cancel out the -3 that was present on the right-hand side, leaving us with just the 3x term. This step demonstrates the importance of using inverse operations to isolate variables. Subtraction and addition are inverse operations, and by adding 3, we are undoing the subtraction of 3 that was originally present in the equation. This process of isolating the variable term is a common strategy in solving various types of algebraic equations, not just linear equations. It is a fundamental technique that is used in more complex equations as well. Therefore, mastering this step is essential for developing strong problem-solving skills in algebra.
Step 4: Solve for x
Finally, to solve for x, we need to isolate x completely. Currently, x is being multiplied by 3. To undo this multiplication, we divide both sides of the equation by 3. This is based on the division property of equality, which states that dividing both sides of an equation by the same non-zero value preserves the equality.
Dividing both sides by 3, we get:
13 / 3 = 3x / 3
Simplifying this, we find:
x = 13/3
Therefore, the solution to the equation 2/3 = (x-1)/5 is x = 13/3. We can express this as an improper fraction (13/3) or as a mixed number (4 1/3). The key to solving for x in this final step is recognizing the inverse operation needed to isolate x. Since x was being multiplied by 3, we used division to undo that multiplication. This process of using inverse operations is a recurring theme in solving algebraic equations and is a fundamental skill to develop. The solution, x = 13/3, represents the value that, when substituted back into the original equation, will make the equation true. This can be verified by plugging 13/3 in for x in the original equation and confirming that both sides are equal. This verification step is a good practice to ensure the accuracy of the solution. Solving for x is the ultimate goal in this process, and by successfully completing this step, we have found the value of the unknown variable that satisfies the given equation.
Verification of the Solution
To ensure the accuracy of our solution, it is always a good practice to verify it. This involves substituting the value we found for x (which is 13/3) back into the original equation and checking if both sides of the equation are equal.
Original equation:
2/3 = (x - 1)/5
Substitute x = 13/3:
2/3 = ((13/3) - 1)/5
Simplify the right-hand side:
2/3 = ((13/3) - (3/3))/5
2/3 = (10/3)/5
2/3 = (10/3) * (1/5)
2/3 = 10/15
2/3 = 2/3
Since both sides of the equation are equal, our solution x = 13/3 is correct. Verification is an essential step in the problem-solving process. It not only confirms the correctness of the solution but also helps in identifying any potential errors made during the solution process. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve the equation. If the left-hand side and the right-hand side of the equation are equal after the substitution, it indicates that our solution is valid. However, if the two sides are not equal, it means that there was an error somewhere in the solution process, and we need to go back and review our steps to identify and correct the error. This process of verification reinforces the understanding of the equation and the solution process. It builds confidence in the solver and ensures that the answer obtained is reliable.
Alternative Methods for Solving the Equation
While we have solved the equation 2/3 = (x-1)/5 using cross-multiplication, there are alternative methods that can be employed. Let's explore one such method:
Method 2: Multiplying Both Sides by the Least Common Multiple (LCM)
Another way to eliminate the fractions in the equation is to multiply both sides by the least common multiple (LCM) of the denominators. In this case, the denominators are 3 and 5. The LCM of 3 and 5 is 15.
Multiply both sides of the equation by 15:
15 * (2/3) = 15 * ((x - 1)/5)
Simplify:
10 = 3(x - 1)
Notice that we have arrived at the same equation as in Step 2 of our previous method. From here, we can proceed with the remaining steps (distribute, isolate the term with x, and solve for x) as before.
This method is particularly useful when dealing with equations that have multiple fractions. Finding the LCM of all the denominators and multiplying both sides by it eliminates all the fractions in one step, simplifying the equation significantly. The concept of the least common multiple (LCM) is fundamental in number theory and is widely used in various mathematical operations, including adding and subtracting fractions. The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. In the context of solving equations with fractions, the LCM of the denominators is used as a common multiple to eliminate the fractions. This method is equivalent to cross-multiplication but provides a more systematic approach, especially when dealing with more complex equations involving multiple fractions. By multiplying both sides of the equation by the LCM, we ensure that all the denominators are canceled out, resulting in an equation with integer coefficients, which is easier to solve. This alternative method reinforces the understanding of different approaches to solving the same problem and provides flexibility in choosing the most efficient method based on the specific equation.
Real-World Applications of Solving for x
Solving for x in equations like 2/3 = (x-1)/5 is not just an abstract mathematical exercise. It has numerous real-world applications across various fields. Let's consider a few examples:
1. Proportional Relationships:
Many real-world scenarios involve proportional relationships. For example, if a recipe calls for 2 cups of flour for every 3 eggs, and you want to use 5 eggs, you can set up a proportion to find out how much flour you need. This proportion would be similar to our equation, where x represents the amount of flour needed.
2. Scaling Recipes:
As mentioned above, solving for x is crucial when scaling recipes. If you want to double a recipe, you need to multiply all the ingredients by 2. If you want to make a fraction of a recipe, you need to multiply all the ingredients by that fraction. Setting up proportions and solving for x allows you to accurately adjust the quantities of ingredients.
3. Unit Conversions:
Unit conversions often involve setting up proportions. For instance, if you want to convert miles to kilometers, you can use the conversion factor 1 mile = 1.609 kilometers. If you know the distance in miles and want to find the distance in kilometers, you can set up a proportion and solve for x.
4. Financial Calculations:
In finance, various calculations involve solving for x. For example, if you want to calculate the interest rate on a loan, you may need to solve an equation for the interest rate. Similarly, if you want to determine the monthly payments on a mortgage, you may need to solve an equation for the monthly payment amount.
5. Physics and Engineering:
In physics and engineering, many equations involve solving for variables. For example, in kinematics, you may need to solve for the velocity, acceleration, or time of an object. In circuit analysis, you may need to solve for the current, voltage, or resistance in a circuit. The ability to solve for variables is, therefore, a fundamental skill in these fields. These real-world applications demonstrate the practical significance of solving for x in equations. The ability to manipulate equations and isolate variables is a powerful tool that can be applied to solve a wide range of problems in various disciplines. The equation 2/3 = (x-1)/5, while seemingly simple, embodies the core principles of algebraic problem-solving that are essential for success in many fields of study and professional endeavors. By mastering the techniques for solving equations like this, individuals can develop critical thinking and analytical skills that are valuable in diverse contexts.
Conclusion
In this comprehensive guide, we have explored the process of solving for x in the equation 2/3 = (x-1)/5. We have delved into the underlying principles of algebraic equations, provided a step-by-step solution, verified the solution, discussed alternative methods, and explored real-world applications. By mastering the techniques presented in this article, you will be well-equipped to tackle similar problems and enhance your problem-solving skills in mathematics and beyond. Solving equations is a fundamental skill that empowers individuals to analyze and solve problems in various aspects of life. The journey of solving 2/3 = (x-1)/5 is not just about finding the value of x; it is about developing a systematic approach to problem-solving, building confidence in one's mathematical abilities, and recognizing the practical relevance of mathematics in the real world. The skills acquired in solving this equation can be transferred to more complex mathematical problems and real-life scenarios, making it a valuable investment of time and effort. The ability to break down a problem into smaller, manageable steps, apply appropriate techniques, and verify the solution is a hallmark of a strong problem-solver. As you continue your mathematical journey, remember that practice and persistence are key to success. With consistent effort, you can master the art of solving equations and unlock the power of mathematics to understand and shape the world around you.
