The problem is to solve the following equation for $x$: $\frac{42}{43}x - \frac{25}{26} = \frac{33}{34}$

1. Problem Description

The problem is to solve the following equation for $x$:

$\frac{42}{43}x - \frac{25}{26} = \frac{33}{34}$

2. Solution Steps

First, we want to isolate the term with $x$. To do this, we add $\frac{25}{26}$ to both sides of the equation:

$\frac{42}{43}x = \frac{33}{34} + \frac{25}{26}$

Next, we need to find a common denominator for $\frac{33}{34}$ and $\frac{25}{26}$.

The least common multiple of $34$ and $26$ is $2 \cdot 17 \cdot 13 = 442$.

So, we rewrite the fractions with the common denominator:

$\frac{33}{34} = \frac{33 \cdot 13}{34 \cdot 13} = \frac{429}{442}$

$\frac{25}{26} = \frac{25 \cdot 17}{26 \cdot 17} = \frac{425}{442}$

Therefore,

$\frac{42}{43}x = \frac{429}{442} + \frac{425}{442}$

$\frac{42}{43}x = \frac{429 + 425}{442} = \frac{854}{442}$

We can simplify the fraction $\frac{854}{442}$ by dividing both the numerator and the denominator by $2$:

$\frac{854}{442} = \frac{427}{221}$

Now we have:

$\frac{42}{43}x = \frac{427}{221}$

To solve for $x$, we multiply both sides of the equation by $\frac{43}{42}$:

$x = \frac{427}{221} \cdot \frac{43}{42}$

$x = \frac{427 \cdot 43}{221 \cdot 42} = \frac{18361}{9282}$

We can simplify the fraction further. Since $221 = 13 \cdot 17$ and $42 = 2 \cdot 3 \cdot 7$, we should check for any common factors between the numerator and the denominator.

Dividing 18361 by 43 gives

4

2

7. Dividing 9282 by 42 gives

2

2

1.

$427 = 31 * 13 + 24$. so 13 is not a factor.

$427 = 17 * 25 + 2$, so 17 is not a factor.

$427 = 2 * 213 + 1$, so 2 is not a factor.

$427 = 3 * 142 + 1$, so 3 is not a factor.

$427 = 7 * 61$, so 7 is not a factor.

$x = \frac{18361}{9282}$

3. Final Answer

$\frac{18361}{9282}$