Solving Linear Equations Step-by-Step Guide

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This comprehensive guide dives deep into the world of linear equations, providing a step-by-step approach to solving for the unknown variable, 'x.' Linear equations are fundamental to mathematics and are encountered in various real-world applications, making it crucial to master the techniques for solving them. We will explore two specific examples in detail, providing clear explanations and strategies that can be applied to a wide range of similar problems.

Understanding Linear Equations

Before we delve into solving the equations, let's establish a solid understanding of what linear equations are. A linear equation is an algebraic equation in which the highest power of the variable is 1. They are called "linear" because when plotted on a graph, they form a straight line. The general form of a linear equation in one variable is:

ax + b = c

Where:

  • x represents the unknown variable we aim to solve for.
  • a is the coefficient of x (a constant that multiplies x).
  • b is a constant term.
  • c is a constant term.

The goal when solving a linear equation is to isolate the variable x on one side of the equation. This means manipulating the equation using algebraic operations until x is by itself, revealing its value.

To effectively solve linear equations, a few core principles of algebra must be understood. These principles allow us to manipulate equations while maintaining their balance and ensuring the solution remains accurate. The most important principles are based on the properties of equality:

  1. Addition Property of Equality: You can add the same number to both sides of an equation without changing its solution. This principle is crucial for eliminating constant terms from one side of the equation.

  2. Subtraction Property of Equality: Similarly, you can subtract the same number from both sides of an equation without affecting the solution. This is another vital tool for isolating the variable.

  3. Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number preserves the equality. This is useful for eliminating fractions or coefficients that are dividing the variable.

  4. Division Property of Equality: Dividing both sides of an equation by the same non-zero number also maintains the equality. This is essential for isolating the variable when it's being multiplied by a coefficient.

  5. Distributive Property: This property states that a(b + c) = ab + ac. It's crucial for simplifying equations where a number is multiplied by a group within parentheses. Applying the distributive property correctly is key to ensuring the equation is in a form that can be easily solved.

  6. Combining Like Terms: Terms with the same variable and exponent can be combined. For example, 3x + 2x can be simplified to 5x. Simplifying by combining like terms makes the equation easier to manipulate and solve.

By applying these principles judiciously, we can transform complex equations into simpler forms, eventually leading to the isolation of the variable and the discovery of its value. Understanding and mastering these principles is the cornerstone of solving any linear equation.

Example 1: Solving -(2/7)x + 6 = 22

Let's tackle the first equation: -(2/7)x + 6 = 22. This equation involves a fraction and requires a few steps to isolate x.

Step 1: Isolate the Term with 'x'

The first step is to isolate the term containing x, which in this case is -(2/7)x. To do this, we need to eliminate the constant term, which is +6. We can achieve this by subtracting 6 from both sides of the equation, applying the Subtraction Property of Equality. This ensures that the equation remains balanced, and we move closer to isolating x.

-(2/7)x + 6 - 6 = 22 - 6

Simplifying both sides, we get:

-(2/7)x = 16

Now, the equation is simplified to a form where the term with x is isolated on one side, making the next steps more straightforward.

Step 2: Eliminate the Fraction

To eliminate the fraction -(2/7), we need to multiply both sides of the equation by the reciprocal of -2/7. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. The reciprocal of -2/7 is -7/2. By multiplying both sides by -7/2, we apply the Multiplication Property of Equality, ensuring the equation remains balanced.

-(2/7)x * (-7/2) = 16 * (-7/2)

On the left side, the fractions cancel out, leaving x by itself. On the right side, we multiply 16 by -7/2.

x = 16 * (-7/2)

Step 3: Simplify

Now, we simplify the right side of the equation. We can multiply 16 by -7 and then divide by 2:

x = (16 * -7) / 2
x = -112 / 2

Finally, we perform the division:

x = -56

Therefore, the solution to the equation -(2/7)x + 6 = 22 is x = -56. This step-by-step approach of isolating the variable, eliminating fractions, and simplifying demonstrates how to solve linear equations involving fractions effectively. Understanding these steps is crucial for tackling more complex equations.

Step 4: Verification of Solution

It is always a good practice to verify the solution to ensure accuracy. To verify x = -56, we substitute this value back into the original equation:

-(2/7)x + 6 = 22
-(2/7)(-56) + 6 = 22

First, we multiply -2/7 by -56:

(-2/7) * (-56) = 16

Now, substitute this result back into the equation:

16 + 6 = 22

Simplifying the left side, we get:

22 = 22

Since both sides of the equation are equal, the solution x = -56 is verified and correct. This verification step is essential to confirm the accuracy of the solution, particularly in more complex equations where errors can easily occur.

Example 2: Solving 50 - 4x = 20x

Now, let's consider the second equation: 50 - 4x = 20x. This equation has x terms on both sides, requiring a slightly different approach to isolate the variable.

Step 1: Combine 'x' Terms

The first step is to combine the x terms on one side of the equation. To do this, we can add 4x to both sides of the equation. This will eliminate the -4x term on the left side and group all x terms on the right side. By applying the Addition Property of Equality, we maintain the balance of the equation.

50 - 4x + 4x = 20x + 4x

Simplifying both sides, we get:

50 = 24x

Now, all the x terms are on one side, making it easier to isolate x.

Step 2: Isolate 'x'

To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 24. This step applies the Division Property of Equality and will leave x by itself on one side of the equation.

50 / 24 = 24x / 24

Simplifying both sides, we get:

x = 50/24

Step 3: Simplify the Fraction

The fraction 50/24 can be simplified by finding the greatest common divisor (GCD) of 50 and 24. The GCD of 50 and 24 is 2. We divide both the numerator and the denominator by 2:

x = (50 / 2) / (24 / 2)
x = 25/12

Therefore, the solution to the equation 50 - 4x = 20x is x = 25/12. This fraction can also be expressed as a mixed number or a decimal, depending on the context or the desired level of precision.

Step 4: Verification of Solution

To verify the solution x = 25/12, we substitute this value back into the original equation:

50 - 4x = 20x
50 - 4(25/12) = 20(25/12)

First, we multiply 4 by 25/12 and 20 by 25/12:

4 * (25/12) = 100/12 = 25/3
20 * (25/12) = 500/12 = 125/3

Now, substitute these results back into the equation:

50 - 25/3 = 125/3

To combine the terms on the left side, we need to express 50 as a fraction with a denominator of 3:

50 = 150/3

Now, substitute this back into the equation:

150/3 - 25/3 = 125/3

Simplifying the left side, we get:

125/3 = 125/3

Since both sides of the equation are equal, the solution x = 25/12 is verified and correct. This thorough verification process ensures that the solution is accurate, reinforcing the importance of this final step in problem-solving.

Key Strategies for Solving Linear Equations

Solving linear equations may appear daunting at first, but with a systematic approach and a solid understanding of algebraic principles, you can confidently tackle even the most complex equations. Here are some key strategies to keep in mind:

  1. Simplify Both Sides: Before attempting to isolate the variable, simplify each side of the equation as much as possible. This includes distributing any multiplication over parentheses, combining like terms, and reducing fractions to their simplest form. Simplifying first makes the subsequent steps easier and reduces the chance of errors. For example, in the equation 3(x + 2) – 5 = 2x + 1, distribute the 3 to get 3x + 6 – 5 = 2x + 1, and then combine like terms to simplify to 3x + 1 = 2x + 1. This step is crucial for managing complex equations effectively.

  2. Isolate the Variable Term: The primary goal is to isolate the term containing the variable on one side of the equation. This usually involves adding or subtracting constants to move them to the opposite side. For example, in the equation 2x + 5 = 11, subtract 5 from both sides to isolate the variable term: 2x = 6. This step is foundational to solving for the variable, so ensure it's done accurately.

  3. Eliminate Fractions and Decimals: Equations containing fractions or decimals can be more challenging to solve. To eliminate fractions, multiply every term in the equation by the least common denominator (LCD). To eliminate decimals, multiply every term by a power of 10 that will convert the decimals to integers. For instance, in the equation x/3 + 1/2 = 1, multiply all terms by the LCD, which is 6, to get 2x + 3 = 6. This strategy simplifies the equation and makes it easier to work with.

  4. Use Inverse Operations: To isolate the variable, use inverse operations. If a number is added to the variable, subtract it from both sides. If a number is multiplied by the variable, divide both sides by it. Inverse operations undo each other and are the key to isolating the variable. In the equation 4x = 16, divide both sides by 4 (the inverse operation of multiplication) to solve for x: x = 4. Mastering inverse operations is essential for solving linear equations.

  5. Maintain Balance: The golden rule of solving equations is to maintain balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This ensures that the equation remains equal and the solution remains valid. This principle is based on the properties of equality and is the cornerstone of algebraic manipulation. It's crucial to adhere to this rule to avoid introducing errors into the solution process.

  6. Check Your Solution: After solving an equation, always check your solution by substituting it back into the original equation. If the equation holds true, your solution is correct. If not, you need to review your steps and find the error. This verification step is crucial for ensuring accuracy and catching any mistakes made during the solving process. It also reinforces your understanding of the equation and the solution.

  7. Stay Organized: Write each step of your solution clearly and neatly. This will help you keep track of your work and make it easier to identify any mistakes. A well-organized approach not only prevents errors but also makes it easier to review and understand the solution process. Use a systematic layout, aligning equal signs and keeping terms in order to maintain clarity.

Conclusion

Solving linear equations is a fundamental skill in mathematics. By understanding the principles of equality, mastering algebraic manipulations, and following a systematic approach, you can confidently solve a wide variety of linear equations. The examples provided in this guide offer a step-by-step illustration of the process, and the key strategies highlighted will serve as valuable tools in your mathematical journey. Remember, practice is key to mastering these skills, so continue to work through problems and challenge yourself to enhance your problem-solving abilities.

These two examples demonstrate the core techniques for solving linear equations. By understanding these steps and practicing regularly, you can confidently solve a wide range of linear equations and build a strong foundation in algebra.