Solving Equations: A Step-by-Step Guide
Hey guys! Let's dive into the world of equation solving. It might seem daunting at first, but trust me, with a little practice, you'll become a pro in no time. We'll break down various types of equations and tackle them step by step. So, grab your pencils and notebooks, and let's get started!
1. Combining Like Terms
Let's kick things off with equations that involve combining like terms. These are your bread and butter, the foundation upon which more complex equation-solving skills are built. Remember, the goal here is to simplify the equation by grouping similar terms together. Think of it as decluttering your math – making everything neat and tidy so you can see the solution clearly.
In this section, we'll be focusing on equations where you need to combine terms with the same variable (like x, y, or a) or constant terms (just numbers). The key here is to remember the basic rules of arithmetic: addition, subtraction, multiplication, and division. Keep your signs straight (a negative sign can be sneaky!), and you'll be golden. We'll use examples to illustrate these concepts, so you can see exactly how it works. We'll start with relatively simple equations and gradually move towards more challenging ones, ensuring you get a solid grasp of the fundamentals. So, let's roll up our sleeves and get to work on these equations – you'll be surprised how quickly you get the hang of it!
1.1 Equation 1: 2x + 5x = 49
Okay, let's start with a classic: 2x + 5x = 49. The first step is to identify the like terms. In this case, we have 2x and 5x. These are like terms because they both contain the variable x. We can combine them by adding their coefficients (the numbers in front of the x). So, 2x + 5x becomes 7x. Now our equation looks like this: 7x = 49. Much simpler, right?
Now, to isolate x, we need to get rid of the 7 that's multiplying it. We do this by performing the inverse operation, which is division. We divide both sides of the equation by 7. Remember, what you do to one side, you must do to the other to keep the equation balanced. So, we have 7x / 7 = 49 / 7. This simplifies to x = 7. And there you have it! We've solved for x. The solution is x = 7. To double-check your work, you can always substitute the value of x back into the original equation. If both sides of the equation are equal, you know you've got the right answer. In this case, 2(7) + 5(7) = 14 + 35 = 49, which confirms our solution.
1.2 Equation 2: 15y - 2y - 2y = 385
Alright, let's tackle another one: 15y - 2y - 2y = 385. Notice how we have multiple terms with the same variable, y. Just like before, we need to combine these like terms. Think of it as adding and subtracting apples – if you have 15 apples, take away 2, and then take away another 2, how many apples do you have left? The same principle applies here.
We start by combining the terms: 15y - 2y - 2y. This simplifies to 11y. So our equation becomes 11y = 385. Now, to isolate y, we need to get rid of the 11 that's multiplying it. We do this by dividing both sides of the equation by 11. This gives us 11y / 11 = 385 / 11. Simplifying, we get y = 35. So the solution is y = 35. To verify, substitute y = 35 back into the original equation: 15(35) - 2(35) - 2(35) = 525 - 70 - 70 = 385. The equation holds true, so we know we've solved it correctly.
1.3 Equation 3: 10a + 2a + 8a = 1544
Let's move on to an equation with the variable a: 10a + 2a + 8a = 1544. Just like the previous examples, our first step is to combine the like terms. We have 10a, 2a, and 8a. These are all like terms because they contain the variable a. To combine them, we simply add their coefficients: 10 + 2 + 8 = 20. So, 10a + 2a + 8a simplifies to 20a. Our equation now looks like this: 20a = 1544.
To isolate a, we need to get rid of the 20 that's multiplying it. We do this by dividing both sides of the equation by 20. This gives us 20a / 20 = 1544 / 20. Simplifying, we get a = 77.2. Therefore, the solution is a = 77.2. Remember, solutions don't always have to be whole numbers! To check our answer, we substitute a = 77.2 back into the original equation: 10(77.2) + 2(77.2) + 8(77.2) = 772 + 154.4 + 617.6 = 1544. The equation holds true, confirming our solution.
1.4 Equation 4: x + x + 2x + 6x = 17120
Time for a slightly bigger number: x + x + 2x + 6x = 17120. Don't let the large number intimidate you; the process is exactly the same. Our first step is, you guessed it, to combine the like terms. We have x, x, 2x, and 6x. Remember that a lone x has an implied coefficient of 1. So, we're essentially adding 1x + 1x + 2x + 6x. Adding the coefficients, we get 1 + 1 + 2 + 6 = 10. Therefore, x + x + 2x + 6x simplifies to 10x. Our equation now looks like this: 10x = 17120.
To isolate x, we divide both sides of the equation by 10: 10x / 10 = 17120 / 10. This simplifies to x = 1712. So, the solution is x = 1712. Let's check our answer by substituting x = 1712 back into the original equation: 1712 + 1712 + 2(1712) + 6(1712) = 1712 + 1712 + 3424 + 10272 = 17120. The equation holds true, confirming that we've solved it correctly. You're getting the hang of this!
2. Equations with Parentheses
Now, let's crank up the difficulty a notch and tackle equations with parentheses. Parentheses might look intimidating, but they're really just a signal to perform an operation in a specific order. In the context of equation solving, parentheses often indicate that we need to use the distributive property. The distributive property is our friend here. It allows us to multiply a number or variable outside the parentheses by each term inside the parentheses. Think of it as sharing the love – the term outside gets multiplied by everyone inside!
This section will focus on equations where you'll need to apply the distributive property to eliminate the parentheses. Once the parentheses are gone, you can then combine like terms and solve for the variable, just like we did in the previous section. The key here is to be meticulous with your signs and remember the order of operations (PEMDAS/BODMAS). We'll walk through several examples, starting with simpler cases and gradually increasing the complexity. This will help you build confidence and master the technique of handling parentheses in equations. So, let's dive in and learn how to tame those parentheses!
2.1 Equation 1: (x + 492) - 798 = 839
Let's start with this equation: (x + 492) - 798 = 839. The first thing we notice is the parentheses around x + 492. However, in this specific case, the parentheses are actually a bit misleading. They don't indicate a multiplication or distribution. Instead, they're simply grouping x + 492 as a single term. So, we can effectively remove them without changing the equation. Our equation now becomes: x + 492 - 798 = 839.
Next, we combine the constant terms on the left side: 492 - 798 = -306. So our equation simplifies to x - 306 = 839. Now, to isolate x, we need to get rid of the -306. We do this by adding 306 to both sides of the equation: x - 306 + 306 = 839 + 306. This simplifies to x = 1145. Therefore, the solution is x = 1145. To check our work, we substitute x = 1145 back into the original equation: (1145 + 492) - 798 = 1637 - 798 = 839. The equation holds true, so we know we've solved it correctly.
2.2 Equation 2: (x - 792) + 297 = 1392
Let's try another one: (x - 792) + 297 = 1392. Similar to the previous equation, the parentheses here are grouping x - 792, but there's no multiplication or distribution involved. We can remove the parentheses without changing the equation: x - 792 + 297 = 1392.
Now, we combine the constant terms on the left side: -792 + 297 = -495. So our equation simplifies to x - 495 = 1392. To isolate x, we need to get rid of the -495. We do this by adding 495 to both sides of the equation: x - 495 + 495 = 1392 + 495. This simplifies to x = 1887. Therefore, the solution is x = 1887. To verify our answer, we substitute x = 1887 back into the original equation: (1887 - 792) + 297 = 1095 + 297 = 1392. The equation holds true, confirming our solution.
2.3 Equation 3: (x - 5342) - 4132 = 9159
Okay, let's move on to: (x - 5342) - 4132 = 9159. Again, the parentheses are simply grouping x - 5342. We can remove them without any distribution: x - 5342 - 4132 = 9159.
Now, we combine the constant terms on the left side: -5342 - 4132 = -9474. So our equation simplifies to x - 9474 = 9159. To isolate x, we add 9474 to both sides of the equation: x - 9474 + 9474 = 9159 + 9474. This simplifies to x = 18633. Therefore, the solution is x = 18633. To check our solution, we substitute x = 18633 back into the original equation: (18633 - 5342) - 4132 = 13291 - 4132 = 9159. The equation holds true, so we've solved it correctly.
2.4 Equation 4: 1952 - (x - 732) = 1713
Now, let's tackle this one: 1952 - (x - 732) = 1713. This equation has a slight twist. We have a negative sign in front of the parentheses. This is crucial because it means we're subtracting the entire expression (x - 732). Think of this negative sign as a -1 being multiplied by the parentheses. So, we need to distribute the -1 across the terms inside the parentheses.
Distributing the -1, we get: -1 * x = -x and -1 * -732 = +732. Our equation now becomes: 1952 - x + 732 = 1713. Next, we combine the constant terms on the left side: 1952 + 732 = 2684. So our equation simplifies to 2684 - x = 1713. Now, we want to isolate x. To do this, we can subtract 2684 from both sides: 2684 - x - 2684 = 1713 - 2684. This gives us -x = -971. But we want x, not -x. To get x, we multiply both sides by -1: -1 * -x = -1 * -971. This simplifies to x = 971. Therefore, the solution is x = 971. To verify our answer, we substitute x = 971 back into the original equation: 1952 - (971 - 732) = 1952 - 239 = 1713. The equation holds true, confirming our solution. Great job!
3. Conclusion
Wow, we've covered a lot! We started with combining like terms and then moved on to tackling equations with parentheses. Remember, the key to solving equations is to take it step by step, stay organized, and double-check your work. Practice makes perfect, so keep at it, and you'll become an equation-solving master in no time! You've got this!
