Solving $16=\frac{7}{6}(4x+18)$ A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into an exciting algebraic equation: $16=\frac{7}{6}(4x+18)$. This equation might seem a bit daunting at first glance, but don't worry, we're going to break it down step by step, making it super easy to understand and solve. Whether you're a student tackling homework, a math aficionado looking for a refresher, or just someone curious about algebra, this guide is for you. We'll explore various strategies, highlight common pitfalls, and ensure you're equipped with the knowledge to conquer similar problems. So, grab your pencils, open your notebooks, and let's embark on this mathematical journey together!

Understanding the Equation

Before we jump into solving, let's first understand what this equation is all about. The equation $16=\frac{7}{6}(4x+18)$ is a linear equation in one variable, which is 'x'. Our goal is to find the value of 'x' that makes this equation true. In simpler terms, we need to figure out what number we can substitute for 'x' so that when we perform the operations on the right side of the equation, we end up with 16. This involves a series of algebraic manipulations, keeping in mind the fundamental principle of maintaining equality. What we do on one side, we must do on the other. This principle ensures that the equation remains balanced throughout our solving process. This equation combines several key concepts in algebra, including the distributive property, fractions, and isolating variables. By mastering this type of equation, you'll be building a solid foundation for more advanced mathematical concepts. So, let's get started and unravel the mystery of 'x'!

Breaking Down the Components

Let's dissect the equation into its key components. On the left side, we have the constant 16. This is our target value – the result we want to achieve after performing operations on the right side. Now, let's turn our attention to the right side of the equation: $\frac{7}{6}(4x+18)$. This side is a bit more complex, involving a fraction and an expression within parentheses. The fraction $\frac{7}{6}$ is a coefficient that will be multiplied by the entire expression inside the parentheses. The expression $(4x+18)$ is a binomial, meaning it consists of two terms: $4x$ and 18. The term $4x$ represents 4 times the variable 'x', and 18 is a constant term. The parentheses indicate that the entire expression $(4x+18)$ needs to be treated as a single unit before multiplying it by $\frac{7}{6}$. Understanding these components is crucial because it dictates the order in which we'll perform our operations. We'll need to address the parentheses first, then deal with the fraction, and finally, isolate 'x' to find its value. By carefully examining each component, we can develop a strategic plan to solve the equation effectively. So, with our components identified, let's move on to the next step: simplifying the equation.

Step-by-Step Solution

Okay, guys, now comes the exciting part – solving the equation! We'll take it one step at a time, making sure each step is crystal clear. Remember, the goal is to isolate 'x' on one side of the equation. Here’s how we’ll do it:

1. Distribute the Fraction

The first step in solving the equation $16=\frac{7}{6}(4x+18)$ is to eliminate the parentheses. To do this, we'll use the distributive property, which states that $a(b + c) = ab + ac$. In our case, 'a' is $\frac{7}{6}$, 'b' is $4x$, and 'c' is 18. So, we need to multiply $\frac{7}{6}$ by both $4x$ and 18. Let's start with $\frac{7}{6} * 4x$. To multiply a fraction by a term with a variable, we multiply the fraction by the coefficient of the variable. So, $\frac{7}{6} * 4x$ becomes $\frac{7 * 4}{6}x$, which simplifies to $\frac{28}{6}x$. We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us $\frac{14}{3}x$. Next, we multiply $\frac{7}{6}$ by 18. This is $\frac{7}{6} * 18$, which is the same as $\frac{7 * 18}{6}$. Multiplying 7 by 18 gives us 126, so we have $\frac{126}{6}$. Now, we divide 126 by 6, which equals 21. So, $\frac{7}{6} * 18$ simplifies to 21. After distributing the fraction, our equation now looks like this: $16 = \frac{14}{3}x + 21$. We've successfully eliminated the parentheses and made the equation a bit more manageable. Now, we can move on to the next step: isolating the term with 'x'.

2. Isolate the Term with 'x'

Now that we've distributed the fraction, our equation is $16 = \frac{14}{3}x + 21$. The next step is to isolate the term containing 'x', which is $\frac{14}{3}x$. To do this, we need to get rid of the constant term on the right side of the equation, which is 21. Remember, the golden rule of algebra is that whatever we do to one side of the equation, we must do to the other. So, to eliminate 21 from the right side, we'll subtract 21 from both sides of the equation. This gives us: $16 - 21 = \frac{14}{3}x + 21 - 21$. On the left side, $16 - 21$ equals -5. On the right side, $21 - 21$ cancels out, leaving us with just $\frac{14}{3}x$. So, our equation now looks like this: $-5 = \frac{14}{3}x$. We've successfully isolated the term with 'x' on one side of the equation. Now, we're just one step away from finding the value of 'x'. The next step is to get rid of the fraction that's multiplying 'x'. Are you ready? Let's move on to the final step!

3. Solve for 'x'

We're almost there, guys! Our equation is currently $-5 = \frac{14}{3}x$. To finally solve for 'x', we need to get rid of the fraction $\frac{14}{3}$ that's multiplying 'x'. The easiest way to do this is to multiply both sides of the equation by the reciprocal of $\frac{14}{3}$. The reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of $\frac{14}{3}$ is $\frac{3}{14}$. Now, let's multiply both sides of the equation by $\frac{3}{14}$: $\frac{3}{14} * -5 = \frac{3}{14} * \frac{14}{3}x$. On the left side, we have $\frac{3}{14} * -5$, which is the same as $\frac{3 * -5}{14}$. This simplifies to $\frac{-15}{14}$. So, the left side is $-\frac{15}{14}$. On the right side, we have $\frac{3}{14} * \frac{14}{3}x$. When we multiply a fraction by its reciprocal, the result is always 1. So, $\frac{3}{14} * \frac{14}{3}$ equals 1. This leaves us with just $1 * x$, which is simply 'x'. So, our equation now looks like this: $-\frac{15}{14} = x$. And there you have it! We've successfully solved for 'x'. The value of 'x' that makes the equation true is $-\frac{15}{14}$. We can also express this as a decimal, which is approximately -1.07. But, as we always say in math, the fractional form is the most accurate representation. Let's take a moment to celebrate our victory! We tackled a seemingly complex equation and emerged triumphant. Now, let's move on to the next section where we'll verify our solution to make sure we didn't make any mistakes along the way.

Verifying the Solution

Alright, champs, we've found our solution for 'x', which is $-\frac{15}{14}$. But, before we declare victory, it's always a good idea to double-check our work and make sure our solution is correct. This process is called verifying the solution, and it's a crucial step in problem-solving. To verify our solution, we'll substitute the value of 'x' back into the original equation and see if both sides of the equation are equal. If they are, then our solution is correct. If not, we'll need to go back and find our mistake. So, let's take our original equation, $16=\frac{7}{6}(4x+18)$, and substitute $-\frac{15}{14}$ for 'x'. This gives us: $16=\frac{7}{6}(4 * -\frac{15}{14}+18)$. Now, we need to simplify the right side of the equation. Let's start with the expression inside the parentheses: $4 * -\frac{15}{14}$. This is the same as $\frac{4 * -15}{14}$, which simplifies to $\frac{-60}{14}$. We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us $-\frac{30}{7}$. So, now we have: $16=\frac{7}{6}(-\frac{30}{7}+18)$. Next, we need to add $-\frac{30}{7}$ and 18. To do this, we need to find a common denominator. The common denominator for 7 and 1 is 7. So, we'll rewrite 18 as a fraction with a denominator of 7: $18 = \frac{18 * 7}{7} = \frac{126}{7}$. Now we can add the fractions: $-\frac{30}{7} + \frac{126}{7} = \frac{-30 + 126}{7} = \frac{96}{7}$. So, our equation now looks like this: $16=\frac{7}{6}(\frac{96}{7})$. Next, we need to multiply $\frac{7}{6}$ by $\frac{96}{7}$. This is the same as $\frac{7 * 96}{6 * 7}$. Notice that we have a 7 in both the numerator and the denominator, so they cancel out. This leaves us with $\frac{96}{6}$. Now, we divide 96 by 6, which equals 16. So, the right side of the equation simplifies to 16. Our equation now looks like this: $16 = 16$. Hooray! Both sides of the equation are equal, which means our solution, $x = -\frac{15}{14}$, is correct. We've successfully verified our solution and can confidently move on to the next section.

Common Mistakes to Avoid

Solving algebraic equations can be tricky, and it's easy to make mistakes if you're not careful. But, don't worry, guys! We're here to help you avoid those common pitfalls. Let's discuss some of the most frequent errors students make when solving equations like $16=\frac{7}{6}(4x+18)$, so you can steer clear of them and ace your math problems.

1. Incorrect Distribution

One of the most common mistakes is distributing the fraction incorrectly. Remember, when you have an expression like $\frac{7}{6}(4x+18)$, you need to multiply $\frac{7}{6}$ by both terms inside the parentheses. Some students mistakenly multiply $\frac{7}{6}$ only by $4x$ or only by 18, which leads to an incorrect equation. To avoid this, always double-check that you've multiplied the fraction by each term inside the parentheses. Write it out step by step if you need to, to ensure you don't miss anything. For example, make sure you calculate both $\frac{7}{6} * 4x$ and $\frac{7}{6} * 18$ separately before combining them in the equation.

2. Order of Operations Errors

Another frequent mistake is not following the correct order of operations. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It tells you the order in which to perform operations. In our equation, we need to address the parentheses first by distributing the fraction. Then, we handle any addition or subtraction before multiplication or division. For instance, some students might try to subtract 21 from both sides of the equation before distributing the fraction, which is incorrect. Always make sure you're following PEMDAS to avoid errors.

3. Sign Errors

Sign errors are also quite common, especially when dealing with negative numbers. For example, when we isolated the term with 'x', we had to subtract 21 from both sides of the equation. If you're not careful, you might make a mistake when subtracting a negative number or adding a positive number. To minimize sign errors, it's a good idea to write out each step clearly and double-check your signs at every stage. If you're unsure, use a number line or other visual aid to help you keep track of the signs.

4. Incorrectly Multiplying by the Reciprocal

When solving for 'x', we multiplied both sides of the equation by the reciprocal of the fraction multiplying 'x'. This is a crucial step, but it's also an area where mistakes can happen. Make sure you correctly identify the reciprocal of the fraction. Remember, the reciprocal is simply flipping the numerator and the denominator. Also, be sure to multiply the entire side of the equation by the reciprocal, not just a part of it. For example, if your equation is $-5 = \frac{14}{3}x$, you need to multiply both -5 and $\frac{14}{3}x$ by $\frac{3}{14}$.

5. Not Verifying the Solution

Finally, one of the biggest mistakes students make is not verifying their solution. Even if you're confident in your steps, it's always a good idea to substitute your solution back into the original equation to make sure it works. This is the best way to catch any errors you might have made along the way. If your solution doesn't check out, go back and carefully review each step to find your mistake. Remember, verifying your solution is like having a safety net – it can save you from losing points on a test or quiz.

By being aware of these common mistakes, you can significantly improve your accuracy when solving algebraic equations. So, keep these pitfalls in mind, practice regularly, and you'll be solving equations like a pro in no time!

Real-World Applications

Now that we've mastered solving the equation $16=\frac{7}{6}(4x+18)$, you might be wondering, where does this stuff actually come in handy in the real world? Well, algebraic equations like this aren't just abstract math problems – they're powerful tools that can be used to model and solve a wide variety of real-world situations. Let's explore some examples of how this type of equation can be applied in practical scenarios.

1. Calculating Costs and Expenses

Imagine you're planning a fundraising event for your school or community organization. You need to figure out how much to charge for tickets to cover your expenses and reach your fundraising goal. Let's say you have fixed costs of $18 (like renting a venue) and variable costs of $4 per attendee (like snacks and drinks). You want to raise a total of $16, and you expect 7/6 of the attendees to actually pay the full ticket price (some might get discounts or free entry). If 'x' represents the number of attendees, the equation $16=\frac{7}{6}(4x+18)$ could represent this situation. Solving this equation would tell you how many attendees you need to achieve your fundraising goal. This type of calculation is essential in budgeting, financial planning, and event management.

2. Determining Proportions and Ratios

Algebraic equations are also useful in situations involving proportions and ratios. For example, let's say you're a chef scaling up a recipe. The original recipe calls for certain amounts of ingredients, but you need to make a larger batch. If you know the ratio of ingredients in the original recipe, you can use an equation to determine the new amounts needed. The equation $16=\frac{7}{6}(4x+18)$ could represent a scenario where you're adjusting ingredient quantities based on a specific proportion. This type of calculation is common in cooking, baking, and other fields where precise measurements are crucial.

3. Solving Geometric Problems

Geometry often involves equations to calculate lengths, areas, and volumes. For example, suppose you have a rectangular garden plot. You know the width of the plot is given by the expression $(4x + 18)$, and you want the area of the garden to be 16 square units. The fraction 7/6 might represent a scaling factor related to the dimensions of the garden. The equation $16=\frac{7}{6}(4x+18)$ could be used to find the value of 'x', which would then allow you to determine the exact dimensions of the garden. This type of problem-solving is common in construction, landscaping, and architecture.

4. Calculating Rates and Speeds

Equations are also fundamental in problems involving rates and speeds. For example, let's say you're planning a road trip. You know the distance you want to travel, and you want to calculate how long it will take at a certain speed. The equation $16=\frac{7}{6}(4x+18)$ could represent a scenario where 'x' is related to the speed of travel, and the other terms represent factors affecting the total time. Solving this equation would help you estimate your travel time and plan your trip effectively. These types of calculations are essential in transportation, logistics, and travel planning.

5. Engineering and Design

In engineering and design, equations are used extensively to model and analyze systems. For instance, the equation $16=\frac{7}{6}(4x+18)$ could represent a simplified model of a structural component, where 'x' represents a design parameter, and the other terms represent constraints and requirements. Solving this equation would help engineers determine the optimal value of 'x' to meet the design specifications. This type of mathematical modeling is crucial in various engineering disciplines, including civil, mechanical, and electrical engineering.

These are just a few examples of the many real-world applications of algebraic equations. By understanding how to solve equations like $16=\frac{7}{6}(4x+18)$, you're equipping yourself with valuable skills that can be applied in a wide range of fields and professions. So, keep practicing, keep exploring, and you'll be amazed at the power of algebra!

Conclusion

Wow, guys, we've covered a lot today! We started with a seemingly complex equation, $16=\frac{7}{6}(4x+18)$, and we broke it down step by step. We learned how to distribute fractions, isolate variables, and solve for 'x'. We even verified our solution to make sure we got it right! Along the way, we discussed common mistakes to avoid and explored some real-world applications of algebraic equations. By now, you should feel confident in your ability to tackle similar problems and apply these concepts in various situations.

The key takeaway here is that algebra isn't just about memorizing formulas and procedures – it's about developing problem-solving skills and critical thinking abilities. The more you practice and the more you explore, the better you'll become at understanding and applying mathematical concepts. So, don't be afraid to challenge yourself, ask questions, and seek out new learning opportunities. Math can be a challenging subject, but it's also incredibly rewarding. The ability to solve equations, analyze data, and make logical arguments is a valuable asset in any field.

Remember, math is like a muscle – the more you use it, the stronger it gets. So, keep exercising your mathematical muscles, and you'll be amazed at what you can achieve. Whether you're pursuing a career in science, technology, engineering, or any other field, a solid foundation in math will serve you well. So, embrace the challenges, celebrate your successes, and never stop learning. Thanks for joining me on this mathematical adventure, and I look forward to exploring more exciting topics with you in the future!