Solving Equations with Fractions: A Comprehensive Guide

When dealing with equations that contain fractions, such as (-frac{x}{6} 13), the key steps involve eliminating the denominator to simplify the expression. This process can seem daunting, but with the right approach, it becomes straightforward. Let's break down the steps and solve the equation systematically.

Step-by-Step Solution

Consider the equation: (-frac{x}{6} 13). The first step is to eliminate the fraction by multiplying both sides of the equation by the denominator, 6.

(-frac{x}{6} cdot 6 13 cdot 6)

This simplifies to:

(-x 78)

At this point, we need to isolate (x). To do this, we multiply both sides of the equation by (-1).

(-x cdot (-1) 78 cdot (-1))

This results in:

(x -78)

Verification

To verify that (x -78) is the correct solution, we can plug it back into the original equation:

(-frac{x}{6} 13 rightarrow -frac{-78}{6} 13)

By simplifying the expression:

(frac{78}{6} 13)

And dividing 78 by 6, we get:

(13 13)

This confirms that our solution is correct.

Additional Examples

Example 1:

-frac{x}{6} 13

( -frac{x}{6} cdot 6 13 cdot 6 ) ( -x 78 ) ( -x cdot (-1) 78 cdot (-1) ) ( x -78 )

Example 2:

(-78 )

This is the solution obtained from the equation ( -frac{x}{6} 13 ).

Multiple Steps

(-x div 6 13 )

( -x div 6 cdot 6 13 cdot 6 ) ( -x 78 ) ( -x cdot (-1) 78 cdot (-1) ) ( x -78 )

Conclusion

Dealing with equations that involve fractions can be challenging, especially when it comes to handling signs carefully. However, with a systematic approach and step-by-step verification, the process becomes much clearer. Always remember to verify your solution by plugging it back into the original equation to ensure accuracy.

Keywords: solving equations, fractions in equations, algebraic equations