Using the solve command effectively
Let's use
solve
to find the roots to a simple quadratic equation. We want to use
x
in our equation, but we have a problem:
x
has been assigned the value 3:
> x;
To clear that assignment, we could type
x := 'x';
which would reset
x
to just "
x
". It's often useful to clear ALL the previous assignments of the current session. That's what the
restart
command does:
> restart;
Now x is just " x ":
> x;
OK, we are ready to solve our quadratic equation:
> sol := solve( x^2 - 7*x + 6 = 0, x );
As we've seen, Maple returns the solutions of the quadratic, arranged in a sequence. Note also that I named the output
sol
. It almost always pays to name the output of a command like
solve
, because then you can refer back to the output by name. The variable
sol
is a sequence of solutions, and we can pick out either solution as
> sol[1]; sol[2];
You could check that these numbers really solve the quadratic by substituting them back into the equation:
>
subs( x = sol[1], x^2 - 7*x + 6 );
subs( x = sol[2], x^2 - 7*x + 6 );
Let's use
solve
to find the roots of a pair of linear equations, first defining the equations and then solving them:
>
eq1 := 2*x - 4*y = 6;
eq2 := x + y = 8;
sols := solve( {eq1, eq2}, {x, y} );
The output of the
solve
command here is a set (it's enclosed in curly braces {}). The set contains two elements, each of which is an equation
> sols[1]; sols[2];
It is possible to use the entire solution in a substitution command.
> subs( sols, x+y );
It's important to note that the solve command has not actually set x or y equal to the stated values---otherwise we'd have assignment operators, := , instead of equal signs.
Here's another example in which the output from
solve
is a bit more complicated. Since I assigned values to
x
and
y
above, I have to clear their values before using them in equations again:
>
restart;
sols:=solve( {x^2 + y^2 = 4, x+y=2}, {x, y} );
In this case there are two solutions. Maple returns them as a sequence, which I named
sols
; each element of the sequence is a set enclosed in curly braces {}, and each set consists of two equations. For example, the first solution is
> sols[1];