Mathematical Solutions and Their Application

Mathematics provides us with a framework to solve equations and prove their validity. This article explores the solution to the equation (frac{x}{52x} - frac{4}{10} 7) and demonstrates the proof for the derived solution. We will use algebraic manipulation and proofs to verify the correctness of the solution.

Problem Statement

Consider the equation:

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Algebraic Solution

We start with the given equation:

(frac{x}{52x} - frac{4}{10} 7)

First, let's denote a common denominator for the terms on the left side. Notice that (52x 2 times 26x) and 10 are in the terms. However, for simplicity, we can proceed by isolating the terms involving (x).

Multiplying both sides by a common denominator or simplifying each term, we get:

(frac{x}{52x} - frac{4}{10} 7 implies frac{x}{52x} - frac{4}{10} 7)

Next, we simplify the equation:

(frac{x}{52x} - frac{4}{10} 7 implies frac{1}{52} - frac{4}{10} 7)

Since the calculation is illustrated with a simplified common denominator, we move on to the next algebraic step:

(frac{2x}{102x} - frac{4}{10} 7 implies frac{2x - 8}{10} 7)

Now, let's solve for (x):

(frac{2x - 8}{10} 7)

Multiplying both sides by 10, we get:

2x - 8 70

Isolating the variable (x), we add 8 to both sides:

(2x 78 implies x frac{78}{2} 39)

Further simplification shows:

(x frac{39}{2} 19.5)

Proof of the Solution

To verify the solution, we substitute (x frac{39}{2}) back into the original equation:

(frac{frac{39}{2}}{52 cdot frac{39}{2}} - frac{4}{10} 7)

(frac{39}{1039} - frac{4}{10} 7)

(frac{70}{10} 7 implies 7 7)

This confirms our solution is correct.

Conclusion

Through meticulous algebraic manipulation and verification, we have shown that the solution to the given equation is (boxed{x frac{39}{2} 19.5}).

Additional Examples

Here are a couple of additional examples to solidify our understanding of equation solving:

Solution 1:

Consider the equation ( frac{x}{5} frac{2x5}{10} 7 ).

Multiplying both sides by a common denominator:

( frac{x}{5} frac{2x5}{10} 7 implies frac{x}{5} frac{2x}{5} 7 )

(implies 2x5 35 implies 2x 30 )

(therefore boxed{x 15})

Solution 2:

Given the equation with a common multiple of 10:

( frac{x}{5} frac{2x5}{10} 7 implies frac{x}{5} frac{2x}{5} - frac{8}{10} 7 )

(implies frac{4x - 8}{10} 7 )

(implies 4x - 8 70 )

(implies 4x 78 )

(implies x frac{78}{4} )

(therefore x frac{39}{2} 19.5 )

Additional Note

We also have the following equation:

Multiply both sides by 10.

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This leads to the final step to verify the equation.