Solving For X: Equations And Step-by-Step Solutions

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Hey guys! Today, we're diving into the world of algebra and tackling the challenge of solving for x in a series of equations. Don't worry if equations seem intimidating at first – we'll break down each one step-by-step, making it super clear and easy to follow. So, grab your pencils, and let's get started on this mathematical adventure!

a) 80 - (30 + 10 - x) = 60

Let's kick things off with our first equation: 80 - (30 + 10 - x) = 60. The key here is to simplify and isolate 'x' on one side of the equation.

  1. Simplify inside the parentheses: First, we'll deal with the expression within the parentheses. We have (30 + 10 - x), which simplifies to (40 - x). So our equation now looks like this: 80 - (40 - x) = 60.

  2. Distribute the negative sign: Next, we need to get rid of the parentheses. Notice the minus sign in front of the parentheses? That means we're actually subtracting the entire expression (40 - x). When we distribute the negative sign, we get: 80 - 40 + x = 60. Remember, subtracting a negative is the same as adding a positive!

  3. Combine like terms: Now, let's simplify the left side of the equation by combining the constant terms: 80 - 40 = 40. This gives us: 40 + x = 60.

  4. Isolate x: Our goal is to get 'x' by itself on one side of the equation. To do this, we need to get rid of the 40 on the left side. We can do this by subtracting 40 from both sides of the equation. This is super important – whatever you do to one side, you have to do to the other to keep the equation balanced! So, we have: 40 + x - 40 = 60 - 40.

  5. Solve for x: Simplifying, we get x = 20. And that's our answer! We've successfully solved for x in the first equation.

So, the solution for the equation 80 - (30 + 10 - x) = 60 is x = 20. Remember, the key to solving these types of equations is to carefully simplify each step and keep the equation balanced.

b) x - (540 - 420) = 70

Now let's move on to the second equation: x - (540 - 420) = 70. This one looks a bit different, but the same principles apply. We still need to simplify and isolate 'x'.

  1. Simplify inside the parentheses: Just like before, let's start by simplifying the expression inside the parentheses. We have (540 - 420), which equals 120. Our equation now looks like this: x - 120 = 70.

  2. Isolate x: To get 'x' by itself, we need to get rid of the -120 on the left side. We can do this by adding 120 to both sides of the equation. Remember, keeping the equation balanced is crucial! So, we have: x - 120 + 120 = 70 + 120.

  3. Solve for x: Simplifying, we get x = 190. Awesome! We've solved for x in the second equation.

The solution for the equation x - (540 - 420) = 70 is x = 190. Notice how simplifying the expression inside the parentheses first made the rest of the problem much easier to handle.

c) (304 - 117 + 27) + x - 15 = 300

Let's tackle the third equation: (304 - 117 + 27) + x - 15 = 300. This one looks a little longer, but don't let that intimidate you. We'll take it step by step.

  1. Simplify inside the parentheses: First, we'll simplify the expression inside the parentheses: (304 - 117 + 27). Let's break it down: 304 - 117 = 187, and then 187 + 27 = 214. So, our equation now looks like this: 214 + x - 15 = 300.

  2. Combine like terms: Next, we'll combine the constant terms on the left side of the equation: 214 - 15 = 199. This gives us: 199 + x = 300.

  3. Isolate x: Now, we need to get 'x' by itself. To do this, we'll subtract 199 from both sides of the equation: 199 + x - 199 = 300 - 199.

  4. Solve for x: Simplifying, we get x = 101. Great job! We've solved for x in the third equation.

The solution for the equation (304 - 117 + 27) + x - 15 = 300 is x = 101. Remember, breaking down complex equations into smaller, manageable steps is key to solving them.

d) (903 - 124 + 31) - x = 400

Moving on to the fourth equation: (903 - 124 + 31) - x = 400. Let's keep practicing our simplification skills!

  1. Simplify inside the parentheses: As usual, we start by simplifying the expression inside the parentheses: (903 - 124 + 31). Let's calculate: 903 - 124 = 779, and then 779 + 31 = 810. So, our equation becomes: 810 - x = 400.

  2. Isolate x: This time, we have '-x' in our equation. To isolate x, we need to get it by itself on one side. There are a couple of ways to do this. One way is to add 'x' to both sides of the equation: 810 - x + x = 400 + x. This simplifies to 810 = 400 + x.

  3. Continue isolating x: Now, we need to get rid of the 400 on the right side. We can do this by subtracting 400 from both sides: 810 - 400 = 400 + x - 400.

  4. Solve for x: Simplifying, we get 410 = x, or x = 410. Fantastic! We've solved for x in the fourth equation.

The solution for the equation (903 - 124 + 31) - x = 400 is x = 410. Remember, dealing with negative variables can be a little tricky, but by following the same principles of simplification and balance, you can solve any equation!

e) 70 + x - 30 + 120 = 250

Let's move on to equation five: 70 + x - 30 + 120 = 250. This one looks a bit more straightforward, so let's jump right in!

  1. Combine like terms: We have several constant terms on the left side of the equation: 70, -30, and 120. Let's combine them: 70 - 30 = 40, and then 40 + 120 = 160. This simplifies our equation to: 160 + x = 250.

  2. Isolate x: To get 'x' by itself, we need to get rid of the 160 on the left side. We can do this by subtracting 160 from both sides of the equation: 160 + x - 160 = 250 - 160.

  3. Solve for x: Simplifying, we get x = 90. Excellent! We've solved for x in the fifth equation.

The solution for the equation 70 + x - 30 + 120 = 250 is x = 90. This equation highlights the importance of combining like terms to make the equation easier to solve.

f) 376 - (25 + 107 - x) = 319

Finally, let's tackle our last equation: 376 - (25 + 107 - x) = 319. We're pros at this now, so let's break it down!

  1. Simplify inside the parentheses: We'll start by simplifying the expression inside the parentheses: (25 + 107 - x). We can simplify the constants: 25 + 107 = 132. So, our expression inside the parentheses becomes (132 - x). The equation now looks like this: 376 - (132 - x) = 319.

  2. Distribute the negative sign: Remember the minus sign in front of the parentheses? We need to distribute it: 376 - 132 + x = 319. Subtracting a negative is the same as adding a positive!

  3. Combine like terms: Now, let's combine the constant terms on the left side: 376 - 132 = 244. This gives us: 244 + x = 319.

  4. Isolate x: To get 'x' by itself, we need to get rid of the 244 on the left side. We can do this by subtracting 244 from both sides of the equation: 244 + x - 244 = 319 - 244.

  5. Solve for x: Simplifying, we get x = 75. We did it! We've solved for x in our final equation.

The solution for the equation 376 - (25 + 107 - x) = 319 is x = 75. This equation reinforces the importance of distributing negative signs correctly and simplifying step by step.

Conclusion: Mastering the Art of Solving for x

Alright, guys! We've made it through all six equations, and you've officially leveled up your algebra skills! Solving for x might seem tricky at first, but by following these key steps, you can conquer any equation that comes your way:

  1. Simplify: Always start by simplifying expressions inside parentheses and combining like terms.
  2. Isolate: Your goal is to get 'x' by itself on one side of the equation. Use inverse operations (addition/subtraction, multiplication/division) to move terms around.
  3. Balance: Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced. This is crucial for maintaining equality.
  4. Check Your Work: Once you've found a solution, plug it back into the original equation to make sure it's correct.

Keep practicing, and you'll become a master at solving for x in no time! Remember, math is like any other skill – the more you practice, the better you'll get. So, keep challenging yourself, and don't be afraid to ask for help when you need it. You've got this!