Triangle Side Analysis: Is 19+X = 7X+1 = 25-X an Equilateral Triangle?

To solve this problem, we will follow these steps:

• Step 1: Calculate when $19 + X = 7X + 1$.
• Step 2: Calculate when $7X + 1 = 25 - X$.
• Step 3: Calculate when $25 - X = 19 + X$.
• Step 4: Check that these conditions are satisfied simultaneously for the triangle to be equilateral.

Step 1: Solve $19 + X = 7X + 1$.

Rearranging terms, we get $19 - 1 = 7X - X$.

This simplifies to $18 = 6X$, giving $X = 3$.

Step 2: Solve $7X + 1 = 25 - X$.

Rearranging terms, we get $7X + X = 25 - 1$.

This simplifies to $8X = 24$, giving $X = 3$.

Step 3: Solve $25 - X = 19 + X$.

Rearranging terms, we get $25 - 19 = X + X$.

This simplifies to $6 = 2X$, giving $X = 3$.

We find that all conditions are satisfied when $X = 3$, and thus all sides are equal at this value.

The side lengths with $X = 3$ are:

• $19 + 3 = 22$
• $7(3) + 1 = 22$
• $25 - 3 = 22$

All sides become equal with $X = 3$, thus the triangle is equilateral.

The solution to the problem is Yes.