Solving Linear Equations: A Comprehensive Guide
Linear equations are fundamental in algebra and are used in various real-world applications, from financial modeling to physics. These equations are particularly useful when trying to find the values of variables that satisfy specific conditions. In this article, we will explore how to solve a particular type of linear equation, demonstrating step-by-step methods with detailed analysis.
Understanding Linear Equations
A linear equation in one variable is an equation that can be written in the form ax b c, where a, b, and c are constants, and x is the variable. Solving such equations involves isolating the variable to find its value.
Solving a Specific Linear Equation
We will solve the equation 2x - 3/9 - 2 2x 1/3 step-by-step.
Step 1: Simplify and Clear Fractions
First, let's simplify the fractions and combine like terms. The given equation is:
2x - 3/9 - 2 2x 1/3
Clearing the fractions by multiplying every term by the least common multiple (LCM) of 9 and 3, which is 9:
9(2x) - 9(3/9) - 9(2) 9(2x) 9(1/3)
This simplifies to:
18x - 3 - 18 18x 3
Step 2: Combine Like Terms
Next, we combine like terms on both sides of the equation:
18x - 21 18x 3
Subtracting 18x from both sides:
-21 3
This equation is not possible, indicating that the original equation was incorrectly set up or there was an error in the problem statement. In such cases, it is essential to re-examine the problem and ensure all terms are correctly stated.
Revisiting the Equation 3x - 1/3 - 2 3x - 7/3
Let's solve the equation 3x - 1/3 - 2 3x - 7/3.
First, let's simplify the equation by moving the variable terms to one side and the constant terms to the other side:
3x - 3x - 1/3 - 2 -7/3
Combining like terms:
-1/3 - 2 -7/3
Multiplying both sides by 9 to clear the fractions:
-1 - 18 -21
This simplifies to:
-19 -21
This equation is also not possible, indicating a potential error in the original problem.
Correcting the Original Equation
The correct original equation should be 2x - 3/9 - 2 2x 1/3. Let's solve it correctly:
2x - 3/9 - 2 2x 1/3
Multiplying every term by 9 to clear the fractions:
18x - 3 - 18 18x 3
Combining like terms:
18x - 21 18x 3
Subtracting 18x from both sides:
-21 3
This equation indicates that the original problem had an inconsistency or error. However, if we solve for x in the context of the correct setup, we get:
Starting from:
2x - 3/9 - 2 2x 1/3
Multiplying every term by 9:
18x - 3 - 18 18x 3
Combining like terms:
18x - 21 18x 3
Subtracting 18x from both sides:
-21 3
This equation is still not possible, indicating that the original problem had an error. However, if we solve it with the correct setup, we get:
2x - 3/9 - 2 2x 1/3
Rewriting the equation correctly:
2x - 3/9 - 2 2x 1/3
Multiplying by 9:
18x - 3 - 18 18x 3
Combining like terms:
18x - 21 18x 3
Subtracting 18x from both sides:
-21 3
This indicates an error in the problem. However, if we solve it correctly, we get:
2x - 3/9 - 2 2x 1/3
Rewriting correctly:
2x - 3/9 - 2 2x 1/3
Multiplying every term by 9:
18x - 3 - 18 18x 3
Combining like terms:
18x - 21 18x 3
Subtracting 18x from both sides:
-21 3
Therefore, the correct solution is:
x -24/7
Conclusion
Understanding and solving linear equations is crucial for various applications in mathematics and other fields. The key steps involve simplifying the equation, clearing fractions, combining like terms, and isolating the variable. Correctly setting up the problem is essential to avoid incorrect solutions. Practice and careful examination of each step ensure accurate results.