# Solve four simultaneous equations? **URL:** \ **Category:** Questions\ **Tags:** algebra, equation-solving, symbolic-computations\ **Created:** [January 15, 2024, 10:11pm UTC](https://community.wolfram.com/t/solve-four-simultaneous-equations/23441 "2024-01-15T22:11:29Z")\ **Posts on this page:** 2\ **Page:** 1
**Author:** ![richardyoung9](https://avatars.discourse-cdn.com/v4/letter/r/848f3c/32.png) [@richardyoung9](https://community.wolfram.com/u/richardyoung9)\ **Post date:** [January 15, 2024, 10:11pm UTC](https://community.wolfram.com/t/solve-four-simultaneous-equations/23441/1 "2024-01-15T22:11:29Z")
I have four equations in four unknowns, a,b,c,d: ``` Sensitivity[a_ ,c_] := (a/(a + c)) * 100 Specificity[b_ , d_] := (d/(b + d)) * 100 PPV[a_ , b_] := (a/(a + b)) * 100 NPV[c_ , d_] := (d/(c + d)) * 100 ``` I would like Mathematica to solve for each of a, b, c, d in terms of Sensitivity, Specificity, PPV and NPV. I don’t know how to do that. Can anyone help? ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [January 15, 2024, 11:41pm UTC](https://community.wolfram.com/t/solve-four-simultaneous-equations/23441/2 "2024-01-15T23:41:13Z")
sensitivity[a\_, c\_] := (a/(a + c))\*100 specificity[b\_, d\_] := (d/(b + d))\*100 pPV[a\_, b\_] := (a/(a + b))\*100 nPV[c\_, d\_] := (d/(c + d))\*100 ``` Solve[{sens == sensitivity[a, c], spec == specificity[b, d], ppv == pPV[a, b], npv == nPV[c, d]}, {a, b, c, d}, MaxExtraConditions -> 1] (* During evaluation of In[932]:= Solve::svars: Equations may not give solutions for all "solve" variables. Out[932]= {{b -> ConditionalExpression[(100 a - a ppv)/ppv, 100 npv ppv - npv ppv sens - npv ppv spec - 100 sens spec + npv sens spec + ppv sens spec == 0], c -> ConditionalExpression[(100 a - a sens)/sens, 100 npv ppv - npv ppv sens - npv ppv spec - 100 sens spec + npv sens spec + ppv sens spec == 0], d -> ConditionalExpression[( a npv (-100 + sens))/((-100 + npv) sens), 100 npv ppv - npv ppv sens - npv ppv spec - 100 sens spec + npv sens spec + ppv sens spec == 0]}} *) ``` There seems to be a needed compatibility condition to make the system consistent. And also a missing equation because as posed the system is homogeneous in `{a,b,c,d}`, which means a normalization is needed or else we get (as we do) a common parametrized factor (`a`, in this case) in the solutions.