Types of Triangles: Solving an equation by multiplying/dividing both sides

To solve the problem, we will perform algebraic manipulation to find $X$.

The triangle gives expressions for sides: $6X$ and $10X - 58$. To find where these are potentially determined equal or prominent in symmetry or division:

• Set the expressions forming these sides equal to each other:

$6X = 10X - 58$

Solve this equation for $X$:

• Subtract $6X$ from both sides:

$0 = 4X - 58$

• Add 58 to both sides:

$58 = 4X$

• Divide both sides by 4 to solve for $X$:

$X = \frac{58}{4}$

Upon simplification:

$X = 14.5$

Therefore, the solution is $X = 14.5$, confirmed as the valid solution satisfying provided problem setup.

Given the expressions $6X$ and $7X + 5$ for two components of a triangle and the lack of any third constraint or piece of information like an actual measure, angle, perimeter relation, or implication about triangle type (isosceles, equilateral, etc.), there is no calculatable conclusion for $X$. Since no method can be consistently or accurately derived from the information provided, it is impossible to definitively solve for $X$.

Therefore, the solution to the problem is that calculating $X$ is Impossible.