User:Djmvfb/cs228

Example 1
Consider the following two equations: $$ \begin{matrix} f(x):\ & 4x - 3y & = & 2 \\ g(x):\ & x + y & = & 10 \end{matrix} $$



What would happen when the coefficient on one variable was changed slightly? For instance: $$ \begin{matrix} f(x):\ & 4x - 3y & = & 2 \\ g_1(x):\ & (1.001)x + y & = & 10 \end{matrix} $$



There is only a very slight change (which isn't even noticeable on the graphs). Whereas the intersection of $$g(x)\ $$ and $$f(x)\ $$ was at $$(4.5714,\ 5.4286)$$, the slightly altered $$g_1(x)\ $$ brings the intersection to $$(4.5695,\ 5.4259)$$. The distance between the two dots is only 0.003302 units, which is generously negligible.