Suppose that I have data $\left\{ (x_i, y_i, z_i ) : i=1, 2, \dots, N\right\} $. I have fitted two linear models: $$ \left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1\\ 1&x_2\\ \vdots&\vdots\\ 1&x_N \end{matrix}\right]\left[\begin{matrix} a_1\\ b_1 \end{matrix}\right] $$ and $$ \left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1&y_1\\ 1&x_2&y_2\\ \vdots&\vdots&\vdots\\ 1&x_N&y_N \end{matrix}\right]\left[\begin{matrix} a_2\\ b_2\\ c_2 \end{matrix}\right].$$ That is, the first model uses $x_i$ values and the second one uses both $x_i$ and $y_i$ values to explain $z_i$.

Now I am considering whether the estimated coefficients $b_1$ and $b_2$ ('slopes') are statistically significant (that is, are the $x_i$:s significant to the model). First of all, I am not sure how to get started with that that problem. Secondly, is it possible that $b_1$ would be significant and $b_2$ would not, or vice versa?