In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$
(I'm working on a algebraically closed field; everything is friendly here).
I need to compute the ideal $I(X)$. I tried to compute generators for the ideal $I(X')\cap I(X'')$, but algebra is the topic in which I'm the worst. I tried to establish the generators that are in both ideals and eventually came to $$ I(X)=\langle xz,yz,xy-ty,x^2-tx\rangle $$ where $\langle f\rangle$ denotes the ideal generated by $f$ in $\mathbb{k}[x,y,z,t]$.
I'm not sure if this is correct and could use some help figuring this out.
Actually you want to prove the following: $$(X,Y)\cap(X-T,Z)=(XZ,YZ,X(X-T),Y(X-T)).$$
Let $f\in(X,Y)\cap(X-T,Z)$. Write $$f=Xf_1+Yf_2=(X-T)g_1+Zg_2\qquad(*)$$ Now sending $X$, $Y$, and $Z$ to $0$ obtain $g_1(0,0,0,T)=0$, so $g_1=Xh_1+Yh_2+Zh_3$. Plugging this in $(*)$ we get $$f=Xf_1+Yf_2=X(X-T)h_1+Y(X-T)h_2+Zk\qquad(**)$$ Sending $X$ and $Y$ to $0$ obtain $k(0,0,Z,T)=0$, so $k=Xk_1+Yk_2$ and you are done.