Algebraic Equations: Solving And Checking
Hey guys! Today, we're diving deep into the fascinating world of algebraic equations. Specifically, we're going to tackle a couple of problems that will not only help you solve equations but also master the crucial step of checking your answers. This skill is super important in math, as it ensures you've got the right solution and understand the underlying principles. We'll be working through two examples: a) 150 + (x - 120) = 870 and b) 380 + (440 - x) = 790. Get ready to boost your algebraic prowess!
Solving Equation A: 150 + (x - 120) = 870
Alright, let's kick things off with our first equation: 150 + (x - 120) = 870. The main goal here is to isolate the variable, 'x'. Think of 'x' as a mystery number we need to uncover. To do this, we need to peel away the numbers and operations surrounding it, one step at a time. First off, we have 150 added to the expression (x - 120). To start isolating the part with 'x', we need to get rid of this 150. The opposite of addition is subtraction, so we'll subtract 150 from both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced. So, we have: 150 + (x - 120) - 150 = 870 - 150. This simplifies to (x - 120) = 720. Now, we've got x with 120 being subtracted from it. To get x by itself, we need to perform the opposite operation of subtraction, which is addition. We'll add 120 to both sides of the equation: (x - 120) + 120 = 720 + 120. And voilà! We get x = 840. So, the solution to our first equation is x = 840. Pretty straightforward, right? Keep these steps in mind as we move on.
Checking Equation A: x = 840
Now comes the really important part: checking our work! This is where we confirm that our solution, x = 840, is indeed correct. To do this, we take our original equation, 150 + (x - 120) = 870, and substitute our value of x back into it. Replace every 'x' with 840. So, the equation becomes 150 + (840 - 120) = 870. Let's work through the left side of the equation. First, we solve the part inside the parentheses: 840 - 120 = 720. Now, substitute that back: 150 + 720 = 870. Finally, perform the addition: 150 + 720 = 870. And look at that – the left side equals the right side (870 = 870). This confirms that our solution x = 840 is absolutely correct! This checking process is your safety net in algebra. It prevents you from submitting an answer that's off, and it solidifies your understanding of how the equation works. Always, always check your solutions, guys!
Solving Equation B: 380 + (440 - x) = 790
Let's move on to our second equation, 380 + (440 - x) = 790. This one looks a little different because the variable 'x' is being subtracted inside the parentheses. But don't sweat it, the strategy is the same: isolate 'x'. First, we need to get the term with 'x' by itself. We have 380 added to (440 - x). So, we'll subtract 380 from both sides of the equation: 380 + (440 - x) - 380 = 790 - 380. This simplifies to (440 - x) = 410. Now, this is where things get a bit tricky. We have 440 minus x equals 410. We want to get 'x' alone. One way to approach this is to realize that if 440 - x = 410, then x must be the difference between 440 and 410. So, we can subtract 410 from 440 to find x: x = 440 - 410. Alternatively, and perhaps more systematically, we can move 'x' to the other side to make it positive. Add 'x' to both sides: 440 - x + x = 410 + x, which gives us 440 = 410 + x. Now, to isolate 'x', subtract 410 from both sides: 440 - 410 = 410 + x - 410. This leaves us with 30 = x, or x = 30. Both methods lead to the same answer, x = 30. The key is to maintain balance and use inverse operations.
Checking Equation B: x = 30
Just like with the first equation, it's time to verify our solution for the second one. We found that x = 30. Let's plug this back into our original equation: 380 + (440 - x) = 790. Substituting x with 30, we get 380 + (440 - 30) = 790. First, let's solve the parentheses: 440 - 30 = 410. Now, substitute this back into the equation: 380 + 410 = 790. Performing the addition on the left side: 380 + 410 = 790. And there you have it – 790 = 790. The equation holds true! This solidifies that our answer, x = 30, is correct. Mastering the checking of algebraic equations is just as vital as solving them. It builds confidence and accuracy in your mathematical work. Don't skip this step, guys; it's your best friend in algebra!
Why Checking Your Answers Matters
So, why do we go through the extra step of checking algebraic solutions? It's more than just a homework requirement; it's a fundamental practice that ensures mathematical integrity. When you solve an equation, you're essentially making a statement about the relationship between numbers and variables. The checking process is how you prove that your statement is true. Firstly, it confirms your accuracy. Math demands precision, and a single misplaced sign or calculation error can lead to a completely wrong answer. Checking allows you to catch these mistakes before they become a problem. Secondly, it reinforces your understanding of algebraic principles. By substituting your solution back into the original equation, you see firsthand how the variable interacts with the constants and operations. This hands-on verification helps solidify the concepts in your mind, making it easier to tackle more complex problems down the line. Think of it like a builder double-checking their measurements before cutting a piece of wood – it saves time and prevents costly errors. Finally, it builds confidence. The more you practice solving and checking, the more assured you become in your abilities. This confidence is invaluable as you progress through higher levels of mathematics and science. So, guys, always make checking a non-negotiable part of your equation-solving routine. It’s the hallmark of a diligent and accurate mathematician!
Conclusion: The Power of Precision in Algebra
In conclusion, mastering algebraic equations involves two equally important skills: solving them accurately and verifying your solutions. We’ve walked through two examples, 150 + (x - 120) = 870 and 380 + (440 - x) = 790, demonstrating that by applying inverse operations systematically, we can isolate the variable and find its value. More importantly, we've emphasized the critical step of checking our answers by substituting the obtained value back into the original equation. This process not only confirms the correctness of our solution but also deepens our comprehension of algebraic concepts. Whether you're just starting with basic linear equations or delving into more complex algebraic structures, the principle remains the same: solve and then check. This habit will serve you incredibly well throughout your mathematical journey, ensuring that your work is not only efficient but also highly accurate. Keep practicing, keep checking, and you'll become a true algebra whiz in no time! Happy solving, everyone!