Solving Equations: Step-by-Step Guide & Solutions

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Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of equations and learn how to solve them like pros. We're going to tackle a set of equations, breaking down each step so you can follow along easily. Plus, we'll show you how to check your answers to make sure they're spot on. Ready? Let's get started!

Why is Solving Equations Important?

Before we jump into the nitty-gritty, let's quickly chat about why solving equations is a crucial skill. Equations are the backbone of so many things in math, science, and even everyday life. Think about it: figuring out how much paint you need for a room, calculating a budget, or even understanding how medicines work – all of these involve equations. Mastering this skill opens doors to deeper understanding and problem-solving abilities in various fields. So, sticking with us through these examples will seriously level up your math game!

When you solve equations, you're essentially uncovering the hidden value of a variable that makes the equation true. This could represent anything from the number of ingredients in a recipe to the speed of a car. The more comfortable you are with equation-solving techniques, the better equipped you'll be to tackle real-world challenges. It's not just about getting the right answer in a textbook; it's about developing a problem-solving mindset that will serve you well in all aspects of life. So, let’s jump into the equations and see how we can crack them together!

Equations and Solutions

1. 4(x + 8) = 44

Okay, let's start with our first equation: 4(x + 8) = 44. The main goal here is to isolate 'x' on one side of the equation. To do this, we'll follow a few key steps:

  • Step 1: Distribute: First, we need to get rid of those parentheses. We do this by distributing the 4 across both terms inside the parentheses. So, 4 times x is 4x, and 4 times 8 is 32. Our equation now looks like this: 4x + 32 = 44
  • Step 2: Isolate the Variable Term: Next, we want to get the term with 'x' by itself. To do that, we need to get rid of the +32. We do this by subtracting 32 from both sides of the equation. Remember, what we do to one side, we have to do to the other to keep things balanced. So, we have: 4x + 32 - 32 = 44 - 32, which simplifies to 4x = 12
  • Step 3: Solve for x: Now, we have 4x = 12. To get 'x' alone, we need to undo the multiplication. We do this by dividing both sides by 4: 4x / 4 = 12 / 4, which gives us x = 3
  • Step 4: Check Your Solution: It's always a good idea to check your answer! We plug x = 3 back into the original equation: 4(3 + 8) = 44. Let's see: 4(11) = 44, and indeed, 44 = 44. So, our solution is correct!

2. 7(x + 8) = 49

Alright, let's tackle the second equation: 7(x + 8) = 49. We're going to use the same steps as before, so it should start feeling familiar!

  • Step 1: Distribute: Multiply the 7 across the terms in the parentheses: 7 * x = 7x, and 7 * 8 = 56. So, the equation becomes: 7x + 56 = 49
  • Step 2: Isolate the Variable Term: We want to get the '7x' term alone, so we need to get rid of the +56. Subtract 56 from both sides: 7x + 56 - 56 = 49 - 56. This simplifies to 7x = -7
  • Step 3: Solve for x: Now, we have 7x = -7. To isolate 'x', divide both sides by 7: 7x / 7 = -7 / 7, which gives us x = -1
  • Step 4: Check Your Solution: Let's plug x = -1 back into the original equation: 7(-1 + 8) = 49. This simplifies to 7(7) = 49, and 49 = 49. Awesome, we got it right!

3. -2(x + 4) = 18

On to the third equation: -2(x + 4) = 18. Don't let the negative sign scare you; we'll handle it the same way!

  • Step 1: Distribute: Multiply the -2 across the terms in the parentheses: -2 * x = -2x, and -2 * 4 = -8. The equation becomes: -2x - 8 = 18
  • Step 2: Isolate the Variable Term: To get '-2x' alone, we need to get rid of the -8. Add 8 to both sides: -2x - 8 + 8 = 18 + 8, which simplifies to -2x = 26
  • Step 3: Solve for x: Now, we have -2x = 26. Divide both sides by -2: -2x / -2 = 26 / -2, which gives us x = -13
  • Step 4: Check Your Solution: Plug x = -13 back into the original equation: -2(-13 + 4) = 18. This simplifies to -2(-9) = 18, and 18 = 18. Perfect!

4. 10(x - 5) = -80

Let's keep the ball rolling with equation number four: 10(x - 5) = -80. We're getting the hang of this, right?

  • Step 1: Distribute: Multiply the 10 across the terms in the parentheses: 10 * x = 10x, and 10 * -5 = -50. So, the equation becomes: 10x - 50 = -80
  • Step 2: Isolate the Variable Term: We need to get '10x' alone, so we'll add 50 to both sides: 10x - 50 + 50 = -80 + 50. This simplifies to 10x = -30
  • Step 3: Solve for x: Now, we have 10x = -30. Divide both sides by 10: 10x / 10 = -30 / 10, which gives us x = -3
  • Step 4: Check Your Solution: Plug x = -3 back into the original equation: 10(-3 - 5) = -80. This simplifies to 10(-8) = -80, and -80 = -80. Nailed it!

5. -5(x - 10) = -35

Equation five is up next: -5(x - 10) = -35. We're getting closer to mastering these equations!

  • Step 1: Distribute: Multiply the -5 across the terms in the parentheses: -5 * x = -5x, and -5 * -10 = 50. Remember, a negative times a negative is a positive! So, the equation becomes: -5x + 50 = -35
  • Step 2: Isolate the Variable Term: To get '-5x' alone, we subtract 50 from both sides: -5x + 50 - 50 = -35 - 50. This simplifies to -5x = -85
  • Step 3: Solve for x: Now, we have -5x = -85. Divide both sides by -5: -5x / -5 = -85 / -5, which gives us x = 17
  • Step 4: Check Your Solution: Plug x = 17 back into the original equation: -5(17 - 10) = -35. This simplifies to -5(7) = -35, and -35 = -35. Fantastic!

6. -9(x - 4) = 81

Let's tackle equation six: -9(x - 4) = 81. We're more than halfway there, guys!

  • Step 1: Distribute: Multiply the -9 across the terms in the parentheses: -9 * x = -9x, and -9 * -4 = 36. The equation becomes: -9x + 36 = 81
  • Step 2: Isolate the Variable Term: To get '-9x' alone, we subtract 36 from both sides: -9x + 36 - 36 = 81 - 36. This simplifies to -9x = 45
  • Step 3: Solve for x: Now, we have -9x = 45. Divide both sides by -9: -9x / -9 = 45 / -9, which gives us x = -5
  • Step 4: Check Your Solution: Plug x = -5 back into the original equation: -9(-5 - 4) = 81. This simplifies to -9(-9) = 81, and 81 = 81. Awesome!

7. 0.4(x - 7) = 18

Time for equation seven: 0.4(x - 7) = 18. Don't let the decimal throw you off; the steps are the same.

  • Step 1: Distribute: Multiply the 0.4 across the terms in the parentheses: 0.4 * x = 0.4x, and 0.4 * -7 = -2.8. The equation becomes: 0.4x - 2.8 = 18
  • Step 2: Isolate the Variable Term: To get '0.4x' alone, we add 2.8 to both sides: 0.4x - 2.8 + 2.8 = 18 + 2.8. This simplifies to 0.4x = 20.8
  • Step 3: Solve for x: Now, we have 0.4x = 20.8. Divide both sides by 0.4: 0. 4x / 0.4 = 20.8 / 0.4, which gives us x = 52
  • Step 4: Check Your Solution: Plug x = 52 back into the original equation: 0.4(52 - 7) = 18. This simplifies to 0.4(45) = 18, and 18 = 18. Great job!

8. -0.25(8 + x) = 14

Here comes equation eight: -0.25(8 + x) = 14. We're on a roll, guys!

  • Step 1: Distribute: Multiply the -0.25 across the terms in the parentheses: -0.25 * 8 = -2, and -0.25 * x = -0.25x. The equation becomes: -2 - 0.25x = 14
  • Step 2: Isolate the Variable Term: To get '-0.25x' alone, we add 2 to both sides: -2 - 0.25x + 2 = 14 + 2. This simplifies to -0.25x = 16
  • Step 3: Solve for x: Now, we have -0.25x = 16. Divide both sides by -0.25: -0.25x / -0.25 = 16 / -0.25, which gives us x = -64
  • Step 4: Check Your Solution: Plug x = -64 back into the original equation: -0.25(8 + (-64)) = 14. This simplifies to -0.25(-56) = 14, and 14 = 14. You're doing amazing!

9. -0.8(10 - x) = 36

Let's tackle equation nine: -0.8(10 - x) = 36. We're getting closer to the finish line!

  • Step 1: Distribute: Multiply the -0.8 across the terms in the parentheses: -0.8 * 10 = -8, and -0.8 * -x = 0.8x. Remember, a negative times a negative is a positive! The equation becomes: -8 + 0.8x = 36
  • Step 2: Isolate the Variable Term: To get '0.8x' alone, we add 8 to both sides: -8 + 0.8x + 8 = 36 + 8. This simplifies to 0.8x = 44
  • Step 3: Solve for x: Now, we have 0.8x = 44. Divide both sides by 0.8: 0.8x / 0.8 = 44 / 0.8, which gives us x = 55
  • Step 4: Check Your Solution: Plug x = 55 back into the original equation: -0.8(10 - 55) = 36. This simplifies to -0.8(-45) = 36, and 36 = 36. Spot on!

10. (1/2)(x - 4) = 5

Alright, equation ten is up: (1/2)(x - 4) = 5. Fractions might look intimidating, but we've got this!

  • Step 1: Distribute: Multiply the (1/2) across the terms in the parentheses: (1/2) * x = (1/2)x, and (1/2) * -4 = -2. The equation becomes: (1/2)x - 2 = 5
  • Step 2: Isolate the Variable Term: To get '(1/2)x' alone, we add 2 to both sides: (1/2)x - 2 + 2 = 5 + 2. This simplifies to (1/2)x = 7
  • Step 3: Solve for x: Now, we have (1/2)x = 7. To get 'x' alone, we can multiply both sides by 2 (which is the reciprocal of 1/2): 2 * (1/2)x = 2 * 7, which gives us x = 14
  • Step 4: Check Your Solution: Plug x = 14 back into the original equation: (1/2)(14 - 4) = 5. This simplifies to (1/2)(10) = 5, and 5 = 5. Awesome job!

11. (4/5)(x + 7) = 20

Equation eleven, here we come: (4/5)(x + 7) = 20. We're rocking these fractional equations!

  • Step 1: Distribute: Multiply the (4/5) across the terms in the parentheses: (4/5) * x = (4/5)x, and (4/5) * 7 = 28/5. The equation becomes: (4/5)x + 28/5 = 20
  • Step 2: Isolate the Variable Term: To get '(4/5)x' alone, we subtract 28/5 from both sides: (4/5)x + 28/5 - 28/5 = 20 - 28/5. To subtract, we need a common denominator, so we rewrite 20 as 100/5: (4/5)x = 100/5 - 28/5, which simplifies to (4/5)x = 72/5
  • Step 3: Solve for x: Now, we have (4/5)x = 72/5. To get 'x' alone, we multiply both sides by the reciprocal of 4/5, which is 5/4: (5/4) * (4/5)x = (5/4) * (72/5), which simplifies to x = 18
  • Step 4: Check Your Solution: Plug x = 18 back into the original equation: (4/5)(18 + 7) = 20. This simplifies to (4/5)(25) = 20, and 20 = 20. You're a pro!

12. -(7/9)(x + 3) = 14

Last but not least, equation twelve: -(7/9)(x + 3) = 14. Let's finish strong!

  • Step 1: Distribute: Multiply the -(7/9) across the terms in the parentheses: -(7/9) * x = -(7/9)x, and -(7/9) * 3 = -7/3. The equation becomes: -(7/9)x - 7/3 = 14
  • Step 2: Isolate the Variable Term: To get '-(7/9)x' alone, we add 7/3 to both sides: -(7/9)x - 7/3 + 7/3 = 14 + 7/3. To add, we need a common denominator, so we rewrite 14 as 42/3: -(7/9)x = 42/3 + 7/3, which simplifies to -(7/9)x = 49/3
  • Step 3: Solve for x: Now, we have -(7/9)x = 49/3. To get 'x' alone, we multiply both sides by the reciprocal of -7/9, which is -9/7: (-9/7) * -(7/9)x = (-9/7) * (49/3), which simplifies to x = -21
  • Step 4: Check Your Solution: Plug x = -21 back into the original equation: -(7/9)(-21 + 3) = 14. This simplifies to -(7/9)(-18) = 14, and 14 = 14. We did it!

Conclusion: You're an Equation-Solving Rockstar!

Wow, guys, we made it through all twelve equations! You've seen how to tackle different types of equations, including those with parentheses, decimals, and fractions. Remember, the key is to follow the steps: distribute, isolate the variable term, solve for the variable, and always, always check your solution. With practice, you'll become more confident and efficient at solving equations. Keep up the great work, and happy equation-solving!