Step-by-Step Solutions Solve Linear Equations

by ADMIN 46 views
Iklan Headers

In the realm of mathematics, solving equations is a fundamental skill. Mastering this skill allows us to unravel the unknowns hidden within mathematical expressions and find the values that make these expressions true. This comprehensive guide will walk you through the process of solving various linear equations, providing detailed, step-by-step solutions to illustrate the techniques involved. We'll break down each equation, explaining the logic behind each step, so you can confidently tackle similar problems in the future. This guide is designed to be your go-to resource for understanding and mastering equation-solving techniques, from basic concepts to more complex applications. So, whether you're a student looking to improve your math skills or simply someone curious about the world of equations, get ready to embark on a journey of mathematical discovery!

1) Solving 2{2(x - 1) + 2} = 18

Let's embark on our journey of solving equations with the first example: 2{2(x - 1) + 2} = 18. This equation may look daunting at first glance, but by systematically applying the order of operations and the properties of equality, we can unravel the mystery and find the value of 'x' that satisfies the equation. The key to successfully solving such equations lies in carefully managing each step, ensuring that we maintain the balance of the equation throughout the process. By understanding the underlying principles and following a clear, methodical approach, even the most complex-looking equations can be tamed. So, let's dive in and explore the step-by-step process of isolating 'x' and revealing its true value. Remember, practice is the key to mastery, and each equation we solve together will strengthen your understanding and build your confidence in tackling future challenges.

  1. Distribute the 2 inside the innermost parentheses:

    2{2(x - 1) + 2} = 18 becomes 2{2x - 2 + 2} = 18

  2. Simplify inside the curly braces:

    2{2x - 2 + 2} = 18 simplifies to 2{2x} = 18

  3. Multiply:

    2{2x} = 18 becomes 4x = 18

  4. Divide both sides by 4 to isolate x:

    4x = 18 leads to x = 18 / 4

  5. Simplify the fraction:

    x = 18 / 4 simplifies to x = 9 / 2 or x = 4.5

Therefore, the solution to the equation 2{2(x - 1) + 2} = 18 is x = 4.5. This detailed walkthrough demonstrates how to systematically break down a complex equation into manageable steps, ultimately leading to the solution. Each step is crucial in maintaining the equation's balance and ensuring the accuracy of the final answer.

2) Solving 6 + 2{x + 3(x + 2)} = 58

Now, let's tackle the second equation: 6 + 2{x + 3(x + 2)} = 58. This equation introduces another layer of complexity with nested parentheses, but fear not! The same principles of order of operations and algebraic manipulation will guide us to the solution. The key is to work from the innermost parentheses outwards, systematically simplifying the expression until we can isolate the variable 'x'. Remember, each step we take must preserve the equality of the equation, ensuring that both sides remain balanced. By carefully following each step and understanding the reasoning behind it, we can confidently navigate this equation and uncover the value of 'x'. So, let's roll up our sleeves and dive into the step-by-step solution process, unraveling the complexities and revealing the hidden value of our variable.

  1. Distribute the 3 inside the parentheses:

    6 + 2{x + 3(x + 2)} = 58 becomes 6 + 2{x + 3x + 6} = 58

  2. Simplify inside the curly braces:

    6 + 2{x + 3x + 6} = 58 simplifies to 6 + 2{4x + 6} = 58

  3. Distribute the 2:

    6 + 2{4x + 6} = 58 becomes 6 + 8x + 12 = 58

  4. Combine like terms on the left side:

    6 + 8x + 12 = 58 simplifies to 8x + 18 = 58

  5. Subtract 18 from both sides:

    8x + 18 = 58 becomes 8x = 40

  6. Divide both sides by 8 to isolate x:

    8x = 40 leads to x = 40 / 8

  7. Simplify:

    x = 40 / 8 simplifies to x = 5

Thus, the solution to the equation 6 + 2{x + 3(x + 2)} = 58 is x = 5. This methodical approach demonstrates the importance of order of operations and careful algebraic manipulation in solving equations with nested parentheses. Each step builds upon the previous one, gradually simplifying the equation until the solution is revealed.

3) Solving 2{3(y - 1) - 2y} = 2

Let's move on to the third equation: 2{3(y - 1) - 2y} = 2. This equation introduces the variable 'y', but the fundamental principles of equation-solving remain the same. We'll systematically apply the order of operations, starting with the innermost parentheses, and use algebraic manipulation to isolate 'y'. The goal is to simplify the equation step by step, ensuring that each operation maintains the balance of the equation. By carefully following the process and understanding the logic behind each step, we can confidently find the value of 'y' that satisfies the equation. So, let's dive in and explore the step-by-step solution, unraveling the equation and revealing the value of our unknown variable.

  1. Distribute the 3 inside the parentheses:

    2{3(y - 1) - 2y} = 2 becomes 2{3y - 3 - 2y} = 2

  2. Simplify inside the curly braces:

    2{3y - 3 - 2y} = 2 simplifies to 2{y - 3} = 2

  3. Distribute the 2:

    2{y - 3} = 2 becomes 2y - 6 = 2

  4. Add 6 to both sides:

    2y - 6 = 2 becomes 2y = 8

  5. Divide both sides by 2 to isolate y:

    2y = 8 leads to y = 8 / 2

  6. Simplify:

    y = 8 / 2 simplifies to y = 4

Therefore, the solution to the equation 2{3(y - 1) - 2y} = 2 is y = 4. This step-by-step solution demonstrates the power of systematic simplification and algebraic manipulation in solving equations. By carefully applying the order of operations and maintaining the balance of the equation, we can confidently arrive at the correct solution.

4) Solving 5{3(x + 2) - 2(x - 1)} = 60

Now, let's tackle the fourth equation: 5{3(x + 2) - 2(x - 1)} = 60. This equation features multiple sets of parentheses and requires careful application of the distributive property and combining like terms. Our strategy remains the same: systematically simplify the equation from the innermost parentheses outwards, ensuring that each step maintains the balance of the equation. By meticulously following the order of operations and carefully performing each calculation, we can unravel the complexities of this equation and reveal the value of 'x'. So, let's embark on the step-by-step solution process, patiently working through each stage to uncover the hidden value of our variable.

  1. Distribute the 3 and the -2 inside the parentheses:

    5{3(x + 2) - 2(x - 1)} = 60 becomes 5{3x + 6 - 2x + 2} = 60

  2. Simplify inside the curly braces:

    5{3x + 6 - 2x + 2} = 60 simplifies to 5{x + 8} = 60

  3. Distribute the 5:

    5{x + 8} = 60 becomes 5x + 40 = 60

  4. Subtract 40 from both sides:

    5x + 40 = 60 becomes 5x = 20

  5. Divide both sides by 5 to isolate x:

    5x = 20 leads to x = 20 / 5

  6. Simplify:

    x = 20 / 5 simplifies to x = 4

Therefore, the solution to the equation 5{3(x + 2) - 2(x - 1)} = 60 is x = 4. This detailed solution highlights the importance of careful distribution and combining like terms when dealing with multiple sets of parentheses. By systematically simplifying the equation, we can arrive at the solution with confidence.

5) Solving 5{2 + 3(x + 2)} = 10

Let's delve into the fifth equation: 5{2 + 3(x + 2)} = 10. This equation, like the others, requires a systematic approach to unravel the value of 'x'. We'll start by simplifying the expression inside the innermost parentheses and work our way outwards, applying the order of operations and maintaining the balance of the equation at each step. By carefully distributing, combining like terms, and performing inverse operations, we can isolate 'x' and reveal its value. So, let's embark on this step-by-step journey, meticulously working through the equation to uncover the solution.

  1. Distribute the 3 inside the parentheses:

    5{2 + 3(x + 2)} = 10 becomes 5{2 + 3x + 6} = 10

  2. Simplify inside the curly braces:

    5{2 + 3x + 6} = 10 simplifies to 5{3x + 8} = 10

  3. Distribute the 5:

    5{3x + 8} = 10 becomes 15x + 40 = 10

  4. Subtract 40 from both sides:

    15x + 40 = 10 becomes 15x = -30

  5. Divide both sides by 15 to isolate x:

    15x = -30 leads to x = -30 / 15

  6. Simplify:

    x = -30 / 15 simplifies to x = -2

Therefore, the solution to the equation 5{2 + 3(x + 2)} = 10 is x = -2. This step-by-step solution demonstrates the importance of careful distribution and simplification, even when dealing with negative numbers. By systematically applying the order of operations and maintaining the balance of the equation, we can confidently arrive at the correct solution.

6) Solving 7x + 5{4 - (x + 1)} = 17

Finally, let's tackle the sixth equation: 7x + 5{4 - (x + 1)} = 17. This equation presents a slight variation with a subtraction within the parentheses, but the core principles of equation-solving remain the same. We'll systematically simplify the equation, paying close attention to the order of operations and the signs of the terms. By carefully distributing, combining like terms, and performing inverse operations, we can isolate 'x' and reveal its value. So, let's embark on this final step-by-step journey, meticulously working through the equation to uncover the solution.

  1. Distribute the negative sign inside the parentheses:

    7x + 5{4 - (x + 1)} = 17 becomes 7x + 5{4 - x - 1} = 17

  2. Simplify inside the curly braces:

    7x + 5{4 - x - 1} = 17 simplifies to 7x + 5{3 - x} = 17

  3. Distribute the 5:

    7x + 5{3 - x} = 17 becomes 7x + 15 - 5x = 17

  4. Combine like terms on the left side:

    7x + 15 - 5x = 17 simplifies to 2x + 15 = 17

  5. Subtract 15 from both sides:

    2x + 15 = 17 becomes 2x = 2

  6. Divide both sides by 2 to isolate x:

    2x = 2 leads to x = 2 / 2

  7. Simplify:

    x = 2 / 2 simplifies to x = 1

Therefore, the solution to the equation 7x + 5{4 - (x + 1)} = 17 is x = 1. This step-by-step solution demonstrates the importance of careful attention to signs and the order of operations when solving equations with parentheses and negative terms. By systematically simplifying the equation, we can confidently arrive at the correct solution.

This comprehensive guide has walked you through the process of solving equations step by step. By understanding the underlying principles and practicing these techniques, you can confidently tackle a wide range of linear equations. Remember, the key is to be systematic, patient, and persistent, and you'll be well on your way to mastering the art of equation solving.