How can I get all solutions of a system of equations with Reduce?

Question

The post What is the difference between Reduce and Solve? discusses the difference between Reduce and Solve. Now, with the system of equations

$$\begin{cases} a^2 + 6b - 7=0,\\ a - \dfrac{1}{b}=0. \end{cases}$$

where $a$, $b$ are real numbers. I tried

Solve[{a^2 + 6 b - 7 == 0 && a - 1/b == 0}, {a, b}, Reals]

{{a -> -3, b -> -(1/3)}, {a -> 1, b -> 1}, {a -> 2, b -> 1/2}}

But when I tried

Reduce[{a^2 + 6 b - 7 == 0 && a - 1/b == 0}, {a, b}, Reals]

I could not get all the solutions. I only got

(a == -3 || a == 1 || a == 2) && b == 1/6 (7 - a^2)

How can I get all the solutions with Reduce?

Answer 1

I'm on 10.0 for Mac OS X x86 (64-bit) (December 4, 2014), just use Backsubstitution -> True while using Reduce,

Solve[{a^2 + 6 b - 7 == 0 && a - 1/b == 0}, {a, b}, Reals]

$\left\{\left\{a\to -3,b\to -\frac{1}{3}\right\},\{a\to 1,b\to 1\},\left\{a\to 2,b\to \frac{1}{2}\right\}\right\}$

Reduce[{a^2 + 6 b - 7 == 0 && a - 1/b == 0}, {a, b}, 
Backsubstitution -> True]

$\left(a=-3\land b=-\frac{1}{3}\right)\lor (a=1\land b=1)\lor \left(a=2\land b=\frac{1}{2}\right)$

You can finde the Options Reduce and Solve with:

Options /@ {Reduce, Solve} // Column

{Backsubstitution -> False, Cubics -> False, GeneratedParameters -> C, Method -> Automatic, Modulus -> 0, Quartics -> False, WorkingPrecision -> ∞}

{Cubics -> True, GeneratedParameters -> C, InverseFunctions -> Automatic, MaxExtraConditions -> 0, Method -> Automatic, Modulus -> 0, Quartics -> True, VerifySolutions -> Automatic, WorkingPrecision -> ∞}

Answer 2

If the output is as shown you can use ToRules:

Reduce[{a^2 + 6 b - 7 == 0 && a - 1/b == 0}, {a, b}, Reals];

{ToRules[%]}

{{a -> -3, b -> 1/6 (7 - a^2)},
 {a -> 1, b -> 1/6 (7 - a^2)},
 {a -> 2, b -> 1/6 (7 - a^2)}}

If you are attempting to find many solutions you may need to increase the System Option "DiscreteSolutionBound":

SetSystemOptions[{"ReduceOptions" -> "DiscreteSolutionBound" -> 100}]

Mathematica enumerates the solutions explicitly only if the number of integer solutions of the system does not exceed the maximum of the p^th power of the value of the system option DiscreteSolutionBound, where p is the dimension of the solution lattice of the equations, and the second element of the value of the system option ExhaustiveSearchMaxPoints.