Solving Linear Equations: A Step-by-Step Guide Exploring the Equation 3x 1 - 5x 12 - 6x - 7
Linear Equations
Linear equations are a fundamental part of algebra, often considered the first step towards more complex mathematical concepts. They represent a relationship between variables, where the highest power of the variable is 1. This simplicity makes them essential for modeling real-world situations, such as financial planning, physics, and engineering.
Algebraic Solutions
Algebra is the branch of mathematics that deals with equations, involving symbols or letters representing numbers or other values. Solving algebraic equations involves isolating the variable on one side of the equation. This process is crucial for finding the value of the unknown variable. Understanding how to solve linear equations is a cornerstone of algebra and is essential for advancing in more advanced mathematical topics.
Solving the Equation 3x 1 - 5x 12 - 6x - 7
To solve the equation 3x 1 - 5x 12 - 6x - 7, we first combine like terms and then isolate the variable. Below is a step-by-step guide to solving this equation.
Premises
The premises for this equation are as follows:
3x 1 - 5x 12 - 6x - 7
Calculations
Step 1: Combine like terms on each side of the equation.
On the left-hand side (LHS):
3x - 5x -2xOn the right-hand side (RHS):
12 - 7 5So, the equation simplifies to:
-2x 1 5 - 6x
Step 2: Move the variable to one side of the equation.
Moving -2x to the right side and 1 to the left side, we get:
1 6x 5 2x
Step 3: Combine the like terms involving the variable on one side.
Subtracting 2x from both sides:
4x 4
Step 4: Divide both sides by the coefficient of the variable to find the solution.
Dividing both sides by 4:
x 1
Therefore, the solution to the equation 3x 1 - 5x 12 - 6x - 7 is x 1.
Verification
To verify the solution, substitute x 1 back into the original equation:
3(1) 1 - 5(1) 12 - 6(1) - 7
3 1 - 5 12 - 6 - 7
-1 -1
The equation is satisfied, confirming that x 1 is the correct solution.
Conclusion
In conclusion, solving linear equations like 3x 1 - 5x 12 - 6x - 7 involves combining like terms, moving the variable to one side, and isolating it to find its value. This method is widely applicable and forms the basis for more complex algebraic problems.
Frequently Asked Questions (FAQs)
What is a linear equation?
A linear equation is an equation in which each term is either a constant or the product of a constant and the first power of a single variable.
How do I solve linear equations?
To solve linear equations, you combine like terms, move all terms involving the variable to one side of the equation, and then isolate the variable to find its value.
What are some practical applications of solving linear equations?
Solving linear equations has practical applications in various fields, such as finance (calculating interest rates), physics (calculating motion and forces), and engineering (designing structures and systems).
Final Thoughts
Solving linear equations is not just a mathematical exercise; it is a fundamental skill with wide-ranging applications. Understanding how to solve them is crucial for anyone looking to delve deeper into algebra or any field that relies on mathematical modeling. By practicing these steps, you can confidently solve any linear equation that comes your way.