Solving For 'a': A Step-by-Step Guide To The Equation
Hey guys! Let's dive into solving for 'a' in the equation: a - 385 - 154 = 784 - a. This might seem a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. We're going to make sure you not only get the answer but also understand why it's the answer. So grab your pencils, and let’s get started!
Understanding the Equation
Okay, so the first thing we need to do is understand what the equation is telling us. We have 'a' minus 385, then minus 154 again, which equals 784 minus 'a'. The key here is to isolate 'a' on one side of the equation. This means we want to get all the 'a' terms on one side and all the numbers on the other side. Remember, the goal is always to simplify until we can clearly see what 'a' equals. To solve equations like a - 385 - 154 = 784 - a, we need to isolate the variable a on one side. This involves combining like terms and performing inverse operations to maintain the equality of the equation. Let's start by simplifying the left side by combining the constants: a - (385 + 154) = 784 - a, which simplifies to a - 539 = 784 - a. Now, we want to get all the a terms on one side and the constants on the other. To do this, we can add a to both sides of the equation: a + a - 539 = 784 - a + a, which simplifies to 2a - 539 = 784. Next, we add 539 to both sides to isolate the term with a: 2a - 539 + 539 = 784 + 539, resulting in 2a = 1323. Finally, we divide both sides by 2 to solve for a: 2a / 2 = 1323 / 2, which gives us a = 661.5. Therefore, the value of a that satisfies the equation is 661.5.
Combining Like Terms
First, let's simplify both sides of the equation. On the left side, we have two numbers being subtracted from 'a': 385 and 154. We can combine these two numbers into one. So, 385 + 154 = 539. This means our equation now looks like this: a - 539 = 784 - a. Combining like terms is a fundamental step in solving algebraic equations. It involves simplifying expressions by grouping together terms that contain the same variable or constant. This process streamlines the equation, making it easier to isolate the variable and find its value. For example, in the equation 3x + 5 - x + 2, we can combine the 3x and -x terms to get 2x, and the 5 and 2 terms to get 7. This simplifies the equation to 2x + 7, making it easier to solve for x. By combining like terms, we reduce the complexity of the equation, making it more manageable and less prone to errors. This step is crucial for efficiently solving algebraic problems and is a cornerstone of algebraic manipulation.
Isolating 'a'
Now comes the fun part – getting 'a' all by itself! Currently, we have 'a' on both sides of the equation, which isn't ideal. To fix this, we need to move one of the 'a's to the other side. Since we have '- a' on the right side, let's add 'a' to both sides of the equation. This will cancel out the '- a' on the right side and give us more 'a's on the left side. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. So, we add 'a' to both sides: a - 539 + a = 784 - a + a. This simplifies to 2a - 539 = 784. Isolating a variable involves rearranging an equation to get the variable alone on one side. This typically requires performing inverse operations to undo any operations acting on the variable. For example, in the equation 3x + 5 = 14, we would first subtract 5 from both sides to isolate the term with x: 3x = 9. Then, we would divide both sides by 3 to solve for x: x = 3. Isolating a variable is a fundamental technique in algebra that allows us to determine the value of the variable that satisfies the equation. It requires careful application of inverse operations and maintaining the balance of the equation to ensure the solution is accurate. This skill is essential for solving a wide range of algebraic problems and is a key component of mathematical proficiency.
Moving the Numbers
We're almost there! Now we need to get rid of the -539 on the left side so that only '2a' is left. To do this, we'll add 539 to both sides of the equation. This will cancel out the -539 on the left side and add to the number on the right side. So, we add 539 to both sides: 2a - 539 + 539 = 784 + 539. This simplifies to 2a = 1323. Moving constants in equations involves isolating the variable term by performing inverse operations on both sides. For example, in the equation 2x - 3 = 7, we want to isolate the term with x, which is 2x. To do this, we add 3 to both sides of the equation: 2x - 3 + 3 = 7 + 3, which simplifies to 2x = 10. Now, the variable term is isolated, and we can proceed to solve for x by dividing both sides by 2. Moving constants is a crucial step in solving algebraic equations, as it allows us to simplify the equation and make it easier to isolate the variable. It requires a careful understanding of inverse operations and maintaining the balance of the equation to ensure the solution is accurate.
Solving for 'a'
Okay, we're in the home stretch! We have 2a = 1323. This means that two times 'a' equals 1323. To find out what 'a' is, we need to divide both sides of the equation by 2. So, we divide both sides by 2: 2a / 2 = 1323 / 2. This simplifies to a = 661.5. And there you have it! We've solved for 'a'! The value of 'a' in the equation is 661.5. To find the value of the variable, we need to divide both sides of the equation by the coefficient of the variable. For example, in the equation 3x = 12, the coefficient of x is 3. To solve for x, we divide both sides of the equation by 3: 3x / 3 = 12 / 3, which simplifies to x = 4. This gives us the value of x that satisfies the equation. Dividing to find the value of the variable is a fundamental step in solving algebraic equations, as it allows us to isolate the variable and determine its value. It requires a careful understanding of division and maintaining the balance of the equation to ensure the solution is accurate. This skill is essential for solving a wide range of algebraic problems and is a key component of mathematical proficiency.
Checking Our Work
It's always a good idea to check our work to make sure we didn't make any mistakes. To do this, we'll plug our answer (a = 661.5) back into the original equation and see if it holds true. So, the original equation is: a - 385 - 154 = 784 - a. Now, let's substitute 661.5 for 'a': 661.5 - 385 - 154 = 784 - 661.5. Let's simplify both sides: 661.5 - 385 - 154 = 122.5 and 784 - 661.5 = 122.5. Since both sides are equal, our answer is correct! Checking solutions is a crucial step in solving mathematical problems. It involves substituting the solution back into the original equation or problem to verify that it satisfies the given conditions. This process helps identify any errors made during the solving process and ensures the accuracy of the solution. For example, if we solve the equation 2x + 3 = 7 and find x = 2, we can check our solution by substituting x = 2 back into the original equation: 2(2) + 3 = 7, which simplifies to 4 + 3 = 7, which is true. This confirms that our solution x = 2 is correct. Checking solutions is a valuable habit to develop, as it helps prevent errors and builds confidence in the correctness of the answers.
Conclusion
And that's how you solve for 'a' in the equation a - 385 - 154 = 784 - a! Remember, the key is to simplify, combine like terms, isolate the variable, and always double-check your work. Solving equations like this becomes easier with practice, so keep at it! You've got this! Keep practicing, and you'll become a pro at solving for unknowns in no time. Equations are the foundation of algebra, and mastering them will open doors to more advanced mathematical concepts. Keep exploring, keep learning, and most importantly, have fun with it!