Consider the following mass distribution where the $x$ - and $y$-coordinates are given in meters: $5.0 \mathrm{~kg}$ at $(0.0,0.0) \mathrm{m}, 2.9 \mathrm{~kg}$ at $(0.0,4.2) \mathrm{m}$, and $4.0 \mathrm{~kg}$ at $(3.5,0.0) \mathrm{m}$. Where should a fourth object of $6.9 \mathrm{~kg}$ be placed so that the center of gravity of the four-object arrangement will be at $(0.0,0.0) \mathrm{m}$ ?
\[
\begin{array}{l}
x=\square m \\
y=\square m
\end{array}
\]

Step 1 ：First, we need to calculate the total mass of the system, which is the sum of the masses of the four objects. The masses are \(5.0 \mathrm{~kg}\), \(2.9 \mathrm{~kg}\), \(4.0 \mathrm{~kg}\), and \(6.9 \mathrm{~kg}\). So, the total mass \(M\) is \(5.0 + 2.9 + 4.0 + 6.9 = 18.8 \mathrm{~kg}\).

Step 2 ：Next, we can substitute the known values into the equations for the center of gravity. The equations are \((m1*x1 + m2*x2 + m3*x3 + m4*x4)/M = 0\) and \((m1*y1 + m2*y2 + m3*y3 + m4*y4)/M = 0\). Substituting the known values, we get \((5.0*0.0 + 2.9*0.0 + 4.0*3.5 + 6.9*x4)/18.8 = 0\) and \((5.0*0.0 + 2.9*4.2 + 4.0*0.0 + 6.9*y4)/18.8 = 0\).

Step 3 ：Solving these equations, we get \(x4 = -2.03\) and \(y4 = -1.77\).

Step 4 ：Final Answer: The fourth object of \(6.9 \mathrm{~kg}\) should be placed at \((-2.03, -1.77) \mathrm{~m}\) so that the center of gravity of the four-object arrangement will be at \((0.0,0.0) \mathrm{~m}\). Therefore, \[ \begin{array}{l} x=\boxed{-2.03} m \ y=\boxed{-1.77} m \end{array} \]