Solving Equations A Detailed Explanation Of 7 + [-5x + (-2x + 3)] = 25 - [(3x + 4) - (4x + 3)]

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Hey guys! Today, we're diving deep into solving a linear equation. We'll take a step-by-step approach to break down the equation: 7 + [-5x + (-2x + 3)] = 25 - [(3x + 4) - (4x + 3)]. This might look intimidating at first, but don't worry! We'll simplify it together, making sure each step is clear and easy to follow. We will also check our answer to make sure everything works out perfectly.

Let's start by taking a closer look at our equation: 7 + [-5x + (-2x + 3)] = 25 - [(3x + 4) - (4x + 3)]. The key to tackling complex equations like this is to break them down into smaller, more manageable parts. We'll simplify each side of the equation separately before bringing them together. This approach helps avoid confusion and makes the entire process much smoother. Remember, math is like building with LEGOs – each piece fits perfectly when you follow the instructions!

Simplifying the Left Side

Okay, let's tackle the left side of the equation first: 7 + [-5x + (-2x + 3)]. Remember the order of operations (PEMDAS/BODMAS)? We need to deal with the parentheses first. Inside the brackets, we have -5x + (-2x + 3). The plus sign in front of the parentheses doesn't change anything, so we can rewrite this as -5x - 2x + 3. Now, combine like terms: -5x - 2x gives us -7x. So, the expression inside the brackets simplifies to -7x + 3. Now, let's bring back the 7 from the beginning: 7 + (-7x + 3). Again, the plus sign doesn't change anything, so we have 7 - 7x + 3. Combine the constants 7 and 3 to get 10. Finally, the left side simplifies to 10 - 7x. See? Not so scary when we take it step by step!

Step-by-Step Breakdown of the Left Side Simplification

  1. Original expression: 7 + [-5x + (-2x + 3)]
  2. Remove inner parentheses: 7 + [-5x - 2x + 3]
  3. Combine like terms inside brackets: 7 + [-7x + 3]
  4. Remove brackets: 7 - 7x + 3
  5. Combine constants: 10 - 7x

Simplifying the Right Side

Now, let's move on to the right side of the equation: 25 - [(3x + 4) - (4x + 3)]. Again, we start with the innermost parentheses. Inside the square brackets, we have (3x + 4) - (4x + 3). This is where things get a little trickier because of the minus sign in front of the second set of parentheses. We need to distribute the negative sign to each term inside the parentheses. This means we change the signs of 4x and 3, making them -4x and -3 respectively. So, (3x + 4) - (4x + 3) becomes 3x + 4 - 4x - 3. Now, combine like terms: 3x - 4x gives us -x, and 4 - 3 gives us 1. So, the expression inside the square brackets simplifies to -x + 1. Don't forget the minus sign in front of the square brackets! We now have 25 - (-x + 1). We need to distribute this negative sign as well, which means changing the signs of -x and 1. This gives us 25 + x - 1. Finally, combine the constants 25 and -1 to get 24. The right side simplifies to 24 + x. We're halfway there!

Step-by-Step Breakdown of the Right Side Simplification

  1. Original expression: 25 - [(3x + 4) - (4x + 3)]
  2. Remove inner parentheses by distributing the negative sign: 25 - [3x + 4 - 4x - 3]
  3. Combine like terms inside brackets: 25 - [-x + 1]
  4. Remove brackets by distributing the negative sign: 25 + x - 1
  5. Combine constants: 24 + x

Combining and Solving for x

Great job, everyone! We've simplified both sides of the equation. Now we have: 10 - 7x = 24 + x. Our goal now is to isolate the variable x. This means getting all the x terms on one side of the equation and all the constants on the other side. Let's start by moving the x term from the right side to the left side. To do this, we subtract x from both sides: 10 - 7x - x = 24 + x - x. This simplifies to 10 - 8x = 24. Next, we want to move the constant 10 from the left side to the right side. We do this by subtracting 10 from both sides: 10 - 8x - 10 = 24 - 10. This simplifies to -8x = 14. Now, we're almost there! To solve for x, we need to divide both sides by -8: -8x / -8 = 14 / -8. This gives us x = -14/8. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us x = -7/4. So, the solution to our equation is x = -7/4.

Step-by-Step Solution for x

  1. Simplified equation: 10 - 7x = 24 + x
  2. Subtract x from both sides: 10 - 7x - x = 24 + x - x which simplifies to 10 - 8x = 24
  3. Subtract 10 from both sides: 10 - 8x - 10 = 24 - 10 which simplifies to -8x = 14
  4. Divide both sides by -8: -8x / -8 = 14 / -8 which simplifies to x = -14/8
  5. Simplify the fraction: x = -7/4

Okay, we've found a solution: x = -7/4. But how do we know it's correct? This is where verification comes in! We need to plug our solution back into the original equation and see if both sides are equal. This process is crucial because it helps us catch any mistakes we might have made along the way. If both sides of the equation are equal after plugging in our solution, then we know we've done everything right. If they're not equal, we need to go back and check our steps to find the error.

The Importance of Verification

Think of verification like double-checking your work on a test. You might feel confident you've got the right answer, but it's always good to make sure. In the same way, plugging our solution back into the original equation gives us peace of mind and confirms that we haven't made any silly mistakes. Trust me, it's worth the extra effort!

Plugging in x = -7/4

Let's get down to business. We'll take our solution, x = -7/4, and substitute it into the original equation: 7 + [-5x + (-2x + 3)] = 25 - [(3x + 4) - (4x + 3)]. This means replacing every instance of x with -7/4. It might look messy, but we'll take it one step at a time. We'll calculate each side separately, just like we did when simplifying the equation.

Step-by-Step Verification Process

  1. Original equation: 7 + [-5x + (-2x + 3)] = 25 - [(3x + 4) - (4x + 3)]
  2. Substitute x = -7/4: 7 + [-5(-7/4) + (-2(-7/4) + 3)] = 25 - [(3(-7/4) + 4) - (4(-7/4) + 3)]
  3. Simplify inside the brackets on the left side:
    • -5(-7/4) = 35/4
    • -2(-7/4) = 14/4 = 7/2
    • 7/2 + 3 = 7/2 + 6/2 = 13/2
    • 35/4 + 13/2 = 35/4 + 26/4 = 61/4
    • 7 + 61/4 = 28/4 + 61/4 = 89/4
  4. Left side simplifies to: 89/4
  5. Simplify inside the brackets on the right side:
    • 3(-7/4) = -21/4
    • -21/4 + 4 = -21/4 + 16/4 = -5/4
    • 4(-7/4) = -7
    • -7 + 3 = -4
    • -5/4 - (-4) = -5/4 + 16/4 = 11/4
    • 25 - 11/4 = 100/4 - 11/4 = 89/4
  6. Right side simplifies to: 89/4

The Verdict

Drumroll, please! We've simplified both sides of the equation after substituting x = -7/4, and guess what? Both sides are equal to 89/4. This means our solution is correct! We've successfully solved the equation and verified our answer. Awesome job, guys!

Phew! We made it through a challenging equation together. We learned how to break down complex expressions, simplify both sides of an equation, solve for the variable, and most importantly, verify our solution. Remember, practice makes perfect! The more equations you solve, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a natural part of learning. Just keep practicing, and you'll become a master equation solver in no time! Keep up the great work, and I'll see you in the next math adventure!

  • Solving Equations
  • Linear Equations
  • Algebra
  • Verification
  • Step-by-Step Solution