Solving the Linear Equation 6x - 7 9x - 17: A Step-by-Step Guide
Linear equations are a fundamental part of algebra and are used extensively in various fields such as physics, engineering, and economics. In this article, we will walk through the process of solving the linear equation 6x - 7 9x - 17. We will break down the solution into clear, manageable steps using algebraic manipulation techniques.
Step 1: Isolating the Variable
Our goal is to isolate the variable x. We start by moving all terms involving x to one side of the equation and the constant terms to the other side. This is done by performing the same operations on both sides of the equation to keep it balanced.
Step 1.1: Initial Equation
Our initial equation is:
6x - 7 9x - 17
Step 1.2: Adding 7 to Both Sides
To move the constant term -7 to the right side, we add 7 to both sides:
6x - 7 7 9x - 17 7
Which simplifies to:
6x 9x - 10
Step 1.3: Subtracting 9x from Both Sides
To move the term 9x to the left side, we subtract 9x from both sides:
6x - 9x 9x - 10 - 9x
Which simplifies to:
-3x -10
Step 2: Simplifying and Solving for x
Now that we have isolated the term involving x on the left side, we can proceed to solve for x by dividing both sides by -3.
Step 2.1: Dividing Both Sides by -3
To isolate x, we divide both sides by -3:
-3x/-3 -10/-3
Which simplifies to:
x 10/3
Therefore, the value of x that makes the equation true is x 10/3.
Verification
To verify our solution, we can substitute x 10/3 back into the original equation:
6x - 7 9x - 17
6(10/3) - 7 9(10/3) - 17
20 - 7 30 - 17
13 13
This confirms that x 10/3 is indeed the correct solution.
Additional Notes
Linear equations like 6x - 7 9x - 17 can also be solved using other algebraic techniques such as moving terms from one side to the other or multiplying/dividing both sides by a constant. Understanding these methods is crucial for students and professionals working with algebra and related fields.
Key Takeaways
- Linear equations are equations in which the highest power of the variable is 1.
- The method of solving linear equations involves isolating the variable by performing the same operations on both sides of the equation.
- Verification is a crucial step to ensure the correctness of the solution.
Conclusion
Mastering the technique of solving linear equations not only helps in solving mathematical problems but also enhances problem-solving skills in various fields. Whether you are a student, a teacher, or a professional, understanding the process of solving 6x - 7 9x - 17 can prove to be a valuable skill.