Algebraic Challenge: Simplify 2-3((-14):(5x/y)-y:(3x×2))

To solve this problem, let's follow these steps:

• Step 1: Simplify the innermost expression within parentheses
• Step 2: Compute the division and subtraction
• Step 3: Multiply the result by $-3$
• Step 4: Add the result to 2

Now, let's work through each step:

Step 1: First, focus on $(-14):\frac{5x}{y}$. The operation suggests dividing $-14$ by $\frac{5x}{y}$, which is equivalent to multiplying by the reciprocal: $-14 \times \frac{y}{5x}$.

Step 2: Simplify the reciprocal multiplication, $-14 \times \frac{y}{5x} = -\frac{14y}{5x}$.

Step 3: Now consider the other term $y:(3x \cdot 2) = \frac{y}{6x}$.

Step 4: Subtract these results: $-\frac{14y}{5x} - \frac{y}{6x}$. To combine fractions, find a common denominator (30x):

$-\frac{14y}{5x} = -\frac{84y}{30x}$ and $\frac{y}{6x} = \frac{5y}{30x}$.

Step 5: The subtraction becomes $-\frac{84y}{30x} - \frac{5y}{30x} = -\frac{89y}{30x}$.

Step 6: Multiply this result by $-3$: $3 \times \left(-\frac{89y}{30x}\right) = \frac{267y}{30x}$. Simplify fractionally to get $\frac{8.9y}{x}$.

Step 7: Add 2 to this result: $2 + \frac{8.9y}{x}$.

Therefore, the solution to the problem is $2 + 8.9\frac{y}{x}$.