The normal sum theorem also works for a linear combination z=ax+by of normal variates x and y. Multiplication of a normal variate by a constant positive number produces another normal variate with proportional changes to the mean and standard deviation so a change in scale will reduce the problem to two cases z = x ± y but the minus sign doesn't affect the joint probability density.
Consequentially the probability density for z in both cases above will be the same after the integration over w irregardless of the sign of y and the variance of z will remain the sum of the two individual variances. For the linear combination var(z) = a2 var(x) + b2 var(y).
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