Solving For W In 3/10 = (w-3)/4 A Step By Step Guide
Solve the equation 3/10 = (w-3)/4 for w. Simplify the answer.
In the realm of mathematics, solving equations is a fundamental skill. This article provides a comprehensive, step-by-step guide on how to solve for w in the equation 3/10 = (w-3)/4. We will explore the underlying principles and techniques involved, ensuring a clear understanding of the process. Whether you are a student learning algebra or simply seeking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems. The key to solving algebraic equations lies in isolating the variable we're interested in – in this case, w. We achieve this by performing operations on both sides of the equation, maintaining the balance and equality. This process often involves cross-multiplication, distribution, and combining like terms. Understanding these fundamental concepts is crucial for mastering algebra and progressing to more advanced mathematical topics. So, let's embark on this journey of solving for w, breaking down each step and clarifying the reasoning behind it. By the end of this guide, you'll not only be able to solve this specific equation but also gain a solid foundation for solving a wide range of algebraic problems. Remember, practice is key to mastery, so work through the examples and try similar problems to reinforce your understanding. Let's begin by diving into the first step: cross-multiplication. This technique allows us to eliminate the fractions and transform the equation into a more manageable form. This initial step is crucial as it sets the stage for the subsequent operations that will lead us to the solution. So, let's move forward and explore the world of equation-solving, one step at a time.
Understanding the Equation
Before diving into the solution, it's crucial to understand the equation we're working with: 3/10 = (w-3)/4. This is a linear equation, which means the variable w is raised to the power of 1. Solving for w involves isolating it on one side of the equation. To achieve this, we'll employ various algebraic techniques. Linear equations are the building blocks of algebra and are encountered in numerous real-world applications, from calculating distances and speeds to modeling financial growth. Mastering the art of solving linear equations is therefore an essential skill. The equation 3/10 = (w-3)/4 represents a proportion, stating that the ratio 3 to 10 is equal to the ratio (w-3) to 4. Understanding this proportional relationship is key to grasping the equation's meaning. We can visualize this proportion as two fractions that are equivalent, highlighting the concept of equality that underpins the equation. The numerator (3) and denominator (10) on the left-hand side represent one ratio, while the numerator (w-3) and denominator (4) on the right-hand side represent the other. The goal is to find the value of w that makes these two ratios equal. This understanding of proportions and ratios is fundamental to many areas of mathematics and science, making the ability to solve such equations highly valuable. So, let's move forward with the solution, keeping in mind the proportional nature of the equation and the need to isolate w to find its value. The next step involves applying the principle of cross-multiplication, a powerful technique for handling equations involving fractions.
Step 1: Cross-Multiplication
The first step in solving this equation is to eliminate the fractions. We achieve this by cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This transforms the equation into a more manageable form without fractions. In our case, we multiply 3 by 4 and 10 by (w-3), resulting in the equation: 3 * 4 = 10 * (w-3). Cross-multiplication is a powerful technique rooted in the fundamental properties of equality. It's essentially a shortcut for multiplying both sides of the equation by the denominators, but it provides a more streamlined approach. Understanding why cross-multiplication works is crucial for building a solid foundation in algebra. By multiplying both sides of the equation by the denominators, we effectively eliminate the fractions, making the equation easier to solve. This technique is widely applicable to equations involving fractions and is a key tool in any algebra student's arsenal. The resulting equation, 3 * 4 = 10 * (w-3), is now free of fractions, making it much simpler to work with. We can now proceed with the next step, which involves simplifying both sides of the equation. This simplification will pave the way for isolating w and finding its value. Remember, each step in the process is designed to bring us closer to the solution, and cross-multiplication is a critical first step in this journey. So, let's move on to the simplification stage, where we'll apply the distributive property and combine like terms to further refine the equation.
Step 2: Simplify Both Sides
Now, let's simplify both sides of the equation we obtained after cross-multiplication: 3 * 4 = 10 * (w-3). On the left side, 3 multiplied by 4 equals 12. On the right side, we need to apply the distributive property. The distributive property states that a(b + c) = ab + ac. So, 10 * (w-3) becomes 10 * w - 10 * 3, which simplifies to 10w - 30. Therefore, our equation now reads: 12 = 10w - 30. Simplifying both sides is a fundamental step in solving algebraic equations. It involves performing the necessary arithmetic operations and applying properties like the distributive property to reduce the equation to its simplest form. This simplification makes it easier to isolate the variable and find its value. The distributive property is a cornerstone of algebra and is used extensively in simplifying expressions and solving equations. It allows us to multiply a number by a sum or difference, distributing the multiplication across the terms inside the parentheses. In our case, applying the distributive property to 10 * (w-3) allows us to eliminate the parentheses and create a simpler expression. The simplified equation, 12 = 10w - 30, is now a two-step equation, meaning we can solve for w in two steps. The next step involves isolating the term with w by adding 30 to both sides of the equation. This will move us closer to our goal of isolating w and finding its value. Remember, each simplification step is crucial for making the equation more manageable and ultimately leading us to the solution. So, let's proceed to the next step and isolate the term containing w.
Step 3: Isolate the Term with w
To isolate the term with w (which is 10w), we need to eliminate the constant term (-30) on the right side of the equation. We do this by adding 30 to both sides of the equation. Adding the same value to both sides maintains the equality of the equation. So, we have: 12 + 30 = 10w - 30 + 30. This simplifies to 42 = 10w. Isolating the term with the variable is a crucial step in solving algebraic equations. It involves strategically adding or subtracting constants to move terms around and group like terms together. The goal is to get the term containing the variable by itself on one side of the equation. This process is based on the fundamental property of equality, which states that adding or subtracting the same value from both sides of an equation preserves the equality. By adding 30 to both sides, we effectively canceled out the -30 on the right side, leaving us with 10w isolated. This step brings us closer to our ultimate goal of finding the value of w. The equation 42 = 10w now shows a clear relationship between the constant 42 and the term 10w. The next step involves dividing both sides of the equation by the coefficient of w (which is 10) to finally isolate w and find its value. Remember, each step is a logical progression towards the solution, and isolating the term with the variable is a key milestone in this journey. So, let's move on to the final step and determine the value of w.
Step 4: Solve for w
Now that we have the equation 42 = 10w, we can solve for w by dividing both sides of the equation by 10. Dividing both sides by the same non-zero value maintains the equality of the equation. So, we have: 42 / 10 = 10w / 10. This simplifies to w = 42/10. Solving for the variable involves performing the inverse operation to isolate it. In this case, since w is multiplied by 10, we divide both sides of the equation by 10 to undo the multiplication. This step directly leads us to the value of w. Dividing both sides by 10 is the final step in isolating w. It effectively cancels out the 10 on the right side, leaving us with w by itself. The result, w = 42/10, is the solution to the equation. However, it's often preferable to simplify the fraction to its lowest terms. The next step involves simplifying the fraction 42/10. This simplification will give us the most concise and easily interpretable value for w. Remember, expressing the solution in its simplest form is a good practice in mathematics. It ensures clarity and makes it easier to compare the solution with other values. So, let's proceed to simplify the fraction and arrive at the final answer.
Step 5: Simplify the Answer
The final step is to simplify the fraction 42/10. Both 42 and 10 are divisible by 2. Dividing both the numerator and the denominator by 2, we get: 42 / 2 = 21 and 10 / 2 = 5. Therefore, w = 21/5. This is the simplified form of the fraction. Simplifying fractions is an essential skill in mathematics. It involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. This process reduces the fraction to its lowest terms, making it easier to work with and understand. In our case, the GCD of 42 and 10 is 2. Dividing both by 2 simplifies the fraction to 21/5. The simplified answer, w = 21/5, is the final solution to the equation. This fraction cannot be simplified further, as 21 and 5 have no common factors other than 1. We can also express this answer as a mixed number: 21 divided by 5 is 4 with a remainder of 1, so w = 4 1/5. Alternatively, we can express the answer as a decimal: 21 divided by 5 is 4.2, so w = 4.2. All three forms of the answer (21/5, 4 1/5, and 4.2) are equivalent and represent the solution to the equation. Choosing the most appropriate form depends on the context of the problem and the desired level of precision. In this case, the simplified fraction 21/5 is a clear and concise representation of the solution. We have now successfully solved for w in the equation 3/10 = (w-3)/4. The step-by-step process involved cross-multiplication, simplification, isolating the term with w, solving for w, and simplifying the answer. This comprehensive guide provides a clear understanding of the techniques involved and equips you with the skills to tackle similar problems in the future.
Final Answer
Therefore, the final answer, in its simplest form, is w = 21/5 or equivalently w = 4.2. We arrived at this solution by systematically applying algebraic principles, ensuring that each step maintained the equality of the equation. This process of solving for a variable is a cornerstone of algebra and has wide-ranging applications in various fields. The solution w = 21/5 represents the value that, when substituted back into the original equation, makes the equation true. We can verify this by substituting 21/5 for w in the equation 3/10 = (w-3)/4 and checking if both sides are equal. This verification step is a good practice to ensure the accuracy of the solution. The decimal representation, w = 4.2, provides an alternative way to express the solution, which may be more convenient in some contexts. Both forms of the answer are mathematically equivalent and represent the same value. We have now successfully navigated the entire process of solving for w, from understanding the equation to arriving at the final answer. This journey has reinforced our understanding of algebraic techniques such as cross-multiplication, simplification, and isolating the variable. The skills acquired in this process will be valuable in tackling more complex mathematical problems in the future. Remember, practice is key to mastering these skills, so continue to work through similar problems and build your confidence in solving algebraic equations. The ability to solve for variables is a fundamental skill in mathematics and a powerful tool for problem-solving in various domains.