Isolate Variable in Equation

Isolate x in the equation a*x^2 + b*x + c == 0.

syms x a b c
eqn = a*x^2 + b*x + c == 0;
xSol = isolate(eqn, x)

xSol =
x == -(b + (b^2 - 4*a*c)^(1/2))/(2*a)

You can use the output of isolate to eliminate the variable from the equation using subs.

Eliminate x from eqn by substituting lhs(xSol) for rhs(xSol).

eqn2 = subs(eqn, lhs(xSol), rhs(xSol))

eqn2 =
c + (b + (b^2 - 4*a*c)^(1/2))^2/(4*a) - (b*(b + (b^2 - 4*a*c)^(1/2)))/(2*a) == 0

Isolate Expression in Equation

Isolate y(t) in the following equation.

syms y(t)
eqn = a*y(t)^2 + b*c == 0;
isolate(eqn, y(t))

ans =
y(t) == ((-b)^(1/2)*c^(1/2))/a^(1/2)

Isolate a*y(t) in the same equation.

isolate(eqn, a*y(t))

ans =
a*y(t) == -(b*c)/y(t)

isolate Returns Simplest Solution

For equations with multiple solutions, isolate returns the simplest solution.

Demonstrate this behavior by isolating x in sin(x) == 0, which has multiple solutions at 0, pi, 3*pi/2, and so on.

isolate(sin(x) == 0, x)

ans =
x == 0

isolate does not consider special cases when returning the solution. Instead, isolate returns a general solution that is not guaranteed to hold for all values of the variables in the equation.

Isolate x in the equation a*x^2/(x-a) == 1. The returned value of x does not hold in the special case a = 0.

syms a x
isolate(a*x^2/(x-a) == 1, x)

ans =
x == ((-(2*a - 1)*(2*a + 1))^(1/2) + 1)/(2*a)

isolate Follows Assumptions on Variables

isolate returns only results that are consistent with the assumptions on the variables in the equation.

First, assume x is negative, and then isolate x in the equation x^4 == 1.

syms x
assume(x < 0)
eqn = x^4 == 1;
isolate(x^4 == 1, x)

ans =
x == -1

Remove the assumption. isolate chooses a different solution to return.

assume(x, 'clear')
isolate(x^4 == 1, x)

ans =
x == 1