Solving Linear Equations Step-by-Step Guide With Examples

Linear equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for success in higher-level mathematics. This article will walk you through the process of solving linear equations with one variable, providing detailed explanations and examples. This section focuses on solving the equation 5x + 9 = 5 + 3x. To solve this equation, our goal is to isolate the variable 'x' on one side of the equation. We can achieve this by performing a series of algebraic operations on both sides of the equation, ensuring that the equality remains balanced. The first step is to gather all terms containing 'x' on one side and the constant terms on the other side. We can subtract 3x from both sides of the equation: 5x + 9 - 3x = 5 + 3x - 3x, which simplifies to 2x + 9 = 5. Next, we need to isolate the term with 'x' by subtracting 9 from both sides: 2x + 9 - 9 = 5 - 9, which simplifies to 2x = -4. Finally, to solve for 'x', we divide both sides by 2: 2x / 2 = -4 / 2, which gives us x = -2. Therefore, the solution to the equation 5x + 9 = 5 + 3x is x = -2. Now, let’s verify our solution by substituting x = -2 back into the original equation: 5(-2) + 9 = 5 + 3(-2). This simplifies to -10 + 9 = 5 - 6, which further simplifies to -1 = -1. Since both sides of the equation are equal, our solution x = -2 is correct. This step-by-step approach ensures accuracy and helps in understanding the underlying principles of solving linear equations. The solution matches option (b) -2. By following these steps, you can confidently solve similar linear equations and reinforce your understanding of algebraic principles. Remember, practice is key to mastering these skills. Keep solving different types of equations to improve your proficiency and problem-solving abilities.

(a) 2 (b) -2 (c) 3 (d) none of these

Solution

The correct answer is (b) -2.

Moving on to the next equation, 4z + 3 = 6 + 2z, we apply the same principles of isolating the variable to find the solution. In this case, the variable is 'z', but the process remains the same as with 'x'. To solve this equation, we first need to gather all terms containing 'z' on one side and the constant terms on the other side. We can start by subtracting 2z from both sides of the equation: 4z + 3 - 2z = 6 + 2z - 2z, which simplifies to 2z + 3 = 6. Next, we need to isolate the term with 'z' by subtracting 3 from both sides: 2z + 3 - 3 = 6 - 3, which simplifies to 2z = 3. Finally, to solve for 'z', we divide both sides by 2: 2z / 2 = 3 / 2, which gives us z = 3/2. Therefore, the solution to the equation 4z + 3 = 6 + 2z is z = 3/2. Now, let's verify our solution by substituting z = 3/2 back into the original equation: 4(3/2) + 3 = 6 + 2(3/2). This simplifies to 6 + 3 = 6 + 3, which further simplifies to 9 = 9. Since both sides of the equation are equal, our solution z = 3/2 is correct. This verification step is crucial to ensure that the solution obtained is accurate and that no errors were made during the solving process. Understanding the steps involved in solving linear equations helps in tackling more complex algebraic problems. The ability to manipulate equations and isolate variables is a fundamental skill in mathematics and is essential for various applications in science and engineering. By consistently practicing and reviewing these techniques, one can build confidence and proficiency in solving linear equations and other algebraic problems. The solution matches option (a) 3/2. This equation further illustrates the importance of careful manipulation and verification in solving linear equations.

(a) 3/2 (b) -3/2 (c) 2 (d) none of these

Solution

The correct answer is (a) 3/2.

The third equation we'll tackle is 2x - 1 = 14 - x. The same principles apply here: we need to isolate 'x' on one side of the equation. This involves moving all terms containing 'x' to one side and all constant terms to the other side through a series of algebraic manipulations. The first step is to add 'x' to both sides of the equation: 2x - 1 + x = 14 - x + x, which simplifies to 3x - 1 = 14. Next, we need to isolate the term with 'x' by adding 1 to both sides: 3x - 1 + 1 = 14 + 1, which simplifies to 3x = 15. Finally, to solve for 'x', we divide both sides by 3: 3x / 3 = 15 / 3, which gives us x = 5. Therefore, the solution to the equation 2x - 1 = 14 - x is x = 5. To ensure our solution is correct, we substitute x = 5 back into the original equation: 2(5) - 1 = 14 - 5. This simplifies to 10 - 1 = 9, which further simplifies to 9 = 9. Since both sides of the equation are equal, our solution x = 5 is correct. This process of verifying the solution is a crucial step in solving equations, as it helps to identify any potential errors made during the algebraic manipulations. By consistently checking our solutions, we can build confidence in our ability to solve equations accurately. The solution matches option (a) 5. This equation provides another opportunity to practice and reinforce the steps involved in solving linear equations, including gathering like terms, isolating the variable, and verifying the solution. Mastery of these steps is essential for success in algebra and related fields. Understanding the logical flow and the importance of each step will enhance your problem-solving skills and make you more proficient in handling algebraic equations.

(a) 5 (b) -9 (c) -5 (d) 9

Solution

The correct answer is (a) 5.

Finally, let's tackle the equation 8x + 4 = 3(x - 1) + 7. This equation introduces a slight variation as it includes parentheses, which require us to apply the distributive property before proceeding with isolating the variable. To solve this equation, we first need to eliminate the parentheses by distributing the 3 across the terms inside the parentheses: 3(x - 1) becomes 3x - 3. So, the equation becomes 8x + 4 = 3x - 3 + 7. Next, we simplify the right side of the equation by combining like terms: -3 + 7 simplifies to 4. Thus, the equation becomes 8x + 4 = 3x + 4. Now, we can proceed with gathering all terms containing 'x' on one side and the constant terms on the other side. We subtract 3x from both sides: 8x + 4 - 3x = 3x + 4 - 3x, which simplifies to 5x + 4 = 4. Next, we subtract 4 from both sides to isolate the term with 'x': 5x + 4 - 4 = 4 - 4, which simplifies to 5x = 0. Finally, to solve for 'x', we divide both sides by 5: 5x / 5 = 0 / 5, which gives us x = 0. Therefore, the solution to the equation 8x + 4 = 3(x - 1) + 7 is x = 0. To verify our solution, we substitute x = 0 back into the original equation: 8(0) + 4 = 3(0 - 1) + 7. This simplifies to 0 + 4 = 3(-1) + 7, which further simplifies to 4 = -3 + 7, and then to 4 = 4. Since both sides of the equation are equal, our solution x = 0 is correct. This equation demonstrates the importance of the distributive property and the order of operations in solving linear equations. By carefully applying these principles, we can accurately solve equations with parentheses and ensure that our solutions are correct. The solution matches option (c) 0. This example reinforces the importance of careful algebraic manipulation and the systematic approach to solving linear equations. Consistent practice and attention to detail are key to mastering these skills.

(a) 1 (b) -1 (c) 0 (d) none of these

Solution

The correct answer is (c) 0.

This comprehensive guide has provided step-by-step solutions to various linear equations, emphasizing the importance of each step in the process. By understanding and practicing these techniques, you can confidently solve linear equations and build a strong foundation in algebra.