13. In \( \triangle M N P \), if \( m \angle M=(4 x-3)^{\circ}, m \angle N=(9 x-6)^{\circ} \), and \( m \angle P=(6 x-1)^{\circ} \), find the value of \( x \) and the measure of each angle. \[ \begin{aligned} x & = \\ m \angle M & = \\ m \angle N & = \\ m \angle P & =\end{aligned} \]

To solve for \( x \) and the measures of each angle in triangle \( MNP \), we start by using the property that the sum of the angles in a triangle is always \( 180^\circ \): \[ m \angle M + m \angle N + m \angle P = 180^\circ \] Substituting the expressions for the angles: \[ (4x - 3) + (9x - 6) + (6x - 1) = 180 \] Combining like terms: \[ 4x + 9x + 6x - 3 - 6 - 1 = 180 \] \[ 19x - 10 = 180 \] Now, isolate \( x \): \[ 19x = 180 + 10 \] \[ 19x = 190 \] \[ x = \frac{190}{19} = 10 \] Now that we have found \( x \), we can find the measures of each angle: \[ m \angle M = 4x - 3 = 4(10) - 3 = 40 - 3 = 37^\circ \] \[ m \angle N = 9x - 6 = 9(10) - 6 = 90 - 6 = 84^\circ \] \[ m \angle P = 6x - 1 = 6(10) - 1 = 60 - 1 = 59^\circ \] So the final answers are: \[ \begin{aligned} x & = 10 \\ m \angle M & = 37^\circ \\ m \angle N & = 84^\circ \\ m \angle P & = 59^\circ \end{aligned} \]