Question: Solve the following heptagon for $x$: Round to two decimals if necessary
$140^\circ$
$(5x -1)^\circ$
$(8x -61)^\circ$
$128^\circ$
$(10x -7)^\circ$
$(7x + 11)^\circ$
$(9x + 25)^\circ$
1. **State the problem:** We are given the interior angles of a heptagon (7-sided polygon) as $140^\circ$, $(5x -1)^\circ$, $(8x -61)^\circ$, $128^\circ$, $(10x -7)^\circ$, $(7x + 11)^\circ$, and $(9x + 25)^\circ$. We need to find the value of $x$.
2. **Formula and rule:** The sum of interior angles of a polygon with $n$ sides is given by:
$$\text{Sum of interior angles} = (n-2) \times 180^\circ$$
For a heptagon, $n=7$, so:
$$\text{Sum} = (7-2) \times 180 = 5 \times 180 = 900^\circ$$
3. **Set up the equation:** The sum of all given angles equals $900^\circ$:
$$140 + (5x -1) + (8x -61) + 128 + (10x -7) + (7x + 11) + (9x + 25) = 900$$
4. **Simplify the equation:** Combine like terms.
Sum of constants:
$$140 - 1 - 61 + 128 - 7 + 11 + 25 = 235$$
Sum of $x$ terms:
$$5x + 8x + 10x + 7x + 9x = 39x$$
So the equation becomes:
$$39x + 235 = 900$$
5. **Solve for $x$:**
Subtract 235 from both sides:
$$39x + \cancel{235} - \cancel{235} = 900 - 235$$
$$39x = 665$$
Divide both sides by 39:
$$x = \frac{665}{39}$$
Use \cancel to show simplification if possible (no common factors besides 1):
$$x = \frac{665}{39}$$
Calculate the decimal value:
$$x \approx 17.05$$
6. **Final answer:**
$$\boxed{x \approx 17.05}$$