Unlocking 'a': Solving The Equation Step-by-Step

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Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for a variable. Specifically, we're tackling the equation 37(a−6)=−2\frac{3}{7}(a-6)=-2. Don't sweat it if equations give you the jitters; we'll break it down into easy-to-digest steps. Our goal is to isolate the variable 'a' and find its value. So, grab your pencils and let's get started. We'll be using some fundamental algebraic principles to unravel this mystery, so stick around and see how it works. Let's start this adventure together, it will be fun, guys!

Understanding the Basics: Equations and Variables

Alright, before we jump into the equation, let's quickly recap what we're dealing with. An equation is a mathematical statement that asserts the equality of two expressions. It's like a seesaw; both sides must balance. In our case, the equation 37(a−6)=−2\frac{3}{7}(a-6)=-2 tells us that the expression on the left-hand side is equal to -2. The goal in solving an equation is to find the value(s) of the variable(s) that make the equation true.

Then, we have variables. A variable is a symbol, usually a letter like 'a', that represents an unknown number. In our equation, 'a' is the variable we need to solve for. Solving for 'a' means figuring out which number, when substituted in place of 'a', makes the equation a true statement. Our task, in essence, is to find the value of 'a' that satisfies the equation 37(a−6)=−2\frac{3}{7}(a-6)=-2. It's like detective work, where 'a' is the hidden treasure, and we need to uncover its value using mathematical tools. Remember, the key is to isolate the variable on one side of the equation.

Now, let's roll up our sleeves and solve the equation. First, we need to get rid of the fraction. To do this, we'll multiply both sides of the equation by the reciprocal of 37\frac{3}{7}, which is 73\frac{7}{3}. This will eliminate the fraction and simplify the equation. After that, we'll deal with the parenthesis and isolate 'a' step by step. I hope you guys are ready, because solving equations can be fun when you understand the logic behind them.

Step-by-Step Solution: Isolating 'a'

Okay, guys, let's get down to the nitty-gritty and solve this equation step by step. We'll go slow and break each step down so you can follow along easily. Remember, our goal is to isolate 'a' on one side of the equation. So, the first step is to get rid of the fraction. Let's do this!

  • Step 1: Eliminate the Fraction. The first thing we want to do is get rid of the fraction 37\frac{3}{7}. To do this, we'll multiply both sides of the equation by the reciprocal of 37\frac{3}{7}, which is 73\frac{7}{3}. Doing this ensures that the equation remains balanced. Applying this to our equation, we get: 73˜⋅37(a−6)=−2â‹…73\~\frac{7}{3} \cdot \frac{3}{7}(a-6) = -2 \cdot \frac{7}{3}

    The left side simplifies because 73⋅37=1\frac{7}{3} \cdot \frac{3}{7} = 1, leaving us with: a−6=−2⋅73a - 6 = -2 \cdot \frac{7}{3}

    Now, let's simplify the right side of the equation: a−6=−143a - 6 = -\frac{14}{3}

  • Step 2: Isolate 'a'. Next, we need to get 'a' by itself. To do this, we'll add 6 to both sides of the equation. This cancels out the -6 on the left side, leaving only 'a'. Adding 6 to both sides gives us: a−6+6=−143+6a - 6 + 6 = -\frac{14}{3} + 6

    This simplifies to: a=−143+6a = -\frac{14}{3} + 6

  • Step 3: Simplify and Find the Value of 'a'. We now have 'a' almost isolated. To finish, we need to add -\frac14}{3} and 6. To do this, we must convert 6 into a fraction with a denominator of 3. Since 6=1836 = \frac{18}{3}, we can rewrite our equation as $a = -\frac{14{3} + \frac{18}{3}$

    Now, we can add the fractions: a=−14+183a = \frac{-14 + 18}{3}

    Which simplifies to: a=43a = \frac{4}{3}

So, the solution to the equation 37(a−6)=−2\frac{3}{7}(a-6)=-2 is a=43a = \frac{4}{3}.

Checking Your Work: Verification

Okay, guys, we've solved the equation and found that a=43a = \frac{4}{3}. But wait, how do we know if our answer is correct? That's where verification comes in. Verification is the process of checking our solution to ensure it satisfies the original equation. It's like double-checking your work to catch any mistakes. Let's substitute our solution, 43\frac{4}{3}, back into the original equation and see if it holds true. If both sides of the equation are equal after the substitution, then our solution is correct. If they are not equal, then we'll need to go back and check our work for any errors. Let's do this!

  • Substitute the value of 'a'. Replace 'a' with 43\frac{4}{3} in the original equation: 37(43−6)=−2\frac{3}{7}(\frac{4}{3} - 6) = -2

  • Simplify the expression inside the parenthesis. First, convert 6 into a fraction with a denominator of 3: 37(43−183)=−2\frac{3}{7}(\frac{4}{3} - \frac{18}{3}) = -2

    Then, perform the subtraction: 37(−143)=−2\frac{3}{7}(\frac{-14}{3}) = -2

  • Multiply the fractions. Multiply 37\frac{3}{7} by −143\frac{-14}{3}: 3⋅−147â‹…3=−2\frac{3 \cdot -14}{7 \cdot 3} = -2

    Simplify the fractions: −4221=−2\frac{-42}{21} = -2

    Which simplifies to: −2=−2-2 = -2

  • Check for equality. Since both sides of the equation are equal (-2 = -2), our solution is verified. This means our value of a=43a = \frac{4}{3} is correct. Congrats, guys, we nailed it! This shows us that we have correctly solved the equation. Verifying our answers is an important step in mathematics, it allows us to ensure the correctness of our solution. Now you know how to solve and verify linear equations. Cool huh?

Conclusion: Mastering the Equation

And there you have it, folks! We've successfully navigated the equation 37(a−6)=−2\frac{3}{7}(a-6)=-2, found the value of 'a', and even verified our answer. Solving for a variable might seem intimidating at first, but with a systematic approach and practice, it becomes much easier. Remember the key steps: isolate the variable, perform the operations on both sides of the equation to maintain balance, and always verify your answer. Math is all about patterns and systems; understanding the steps will allow you to solve even the most complex equations.

In summary, here's what we did:

  1. We started by understanding the basics of equations and variables.
  2. We eliminated the fraction by multiplying both sides of the equation by the reciprocal of 37\frac{3}{7}.
  3. We isolated 'a' by adding 6 to both sides of the equation.
  4. We simplified the expression and found the value of 'a' to be 43\frac{4}{3}.
  5. We verified our answer by substituting the value back into the original equation, confirming that our solution was correct.

We started with a complex-looking equation and broke it down into easy, manageable steps. That is the beauty of mathematics. Remember to practice regularly, because the more you practice, the more comfortable you'll become with solving equations. You will gradually improve and learn new strategies for solving different types of equations. Keep up the excellent work, and always ask questions. Good luck and happy solving!