Solving for an Unknown Number in Algebraic Equations: A Step-by-Step Guide
Algebra is a fundamental part of mathematics, and understanding how to solve algebraic equations is crucial for students and professionals alike. One common type of problem involves determining an unknown number based on a given expression. In this article, we'll explore how to solve such problems systematically, using a specific example where 1 is subtracted from eight times a certain number to get a result of 15. This step-by-step approach will help you grasp the underlying concepts and techniques.
Introduction to Algebraic Equations
Before delving into solving the specific problem, let's briefly review what algebraic equations are. An algebraic equation is a statement that two expressions are equal. In this case, we have an equation where 1 is subtracted from eight times a certain number, and the result is 15. We will denote this unknown number as ( x ).
Solving the Equation Step-by-Step
Let's set up the equation based on the information given:
8x - 1 15
Our goal is to solve for ( x ). We'll do this in several steps:
Isolate the term involving ( x ): Add 1 to both sides of the equation to eliminate the constant term on the left side. Combine like terms: This simplifies the left side of the equation. Solve for ( x ): Divide both sides of the equation by the coefficient of ( x ).Let's go through each step in detail:
Step 1: Add 1 to Both Sides
8x - 1 1 15 1
8x 16
By adding 1 to both sides, we eliminate the -1 on the left, leaving us with a simple expression involving ( x ).
Step 2: Combine Like Terms (if necessary)
In this case, there are no more terms to combine on the left side.
Step 3: Solve for ( x )
8x 16
x frac{16}{8}
x 2
Finally, we divide both sides by 8 to isolate ( x ), giving us the solution ( x 2 ).
Common Mistakes and Clarifications
To ensure correct understanding, let's address some common mistakes and clarify any confusions.
Example 1: 8x - 1 15
Following the steps above, we can easily solve this equation and verify the solution:
8x - 1 15
8x 16
x 2
The solution is confirmed to be ( x 2 ).
Example 2: 8x - 1 15 (Alternative Approach)
In this example, we follow similar steps but use a different approach:
8x - 1 15
8x 15 1
8x 16
x frac{16}{8}
x 2
The solution remains ( x 2 ).
Example 3: 8m - 1 15
If the unknown number is ( m ) instead of ( x ), the process is identical:
8m - 1 15
8m 16
m frac{16}{8}
m 2
Again, the solution is ( m 2 ).
Conclusion
Solving algebraic equations can be straightforward with the right approach. By following the steps of isolating the variable term, combining like terms, and then solving for the variable, you can confidently find the unknown number in a given equation. Practice is key, and using specific examples helps in solidifying the concepts and techniques.