Solutions of an EquationIf we substitute x by -3 in the equation 2x + 8 = -2x - 4, we obtain
left side: 2x + 8 = 2(-3) + 8 = -6 + 8 = 2
right side: -2x - 4 = -2(-3) - 4 = 6 - 4 = 2
Since a substitution of x = - 3 in the equation gives a true statement 2 = 2, we call -3 the solution or root of the given equation 2x + 8 = -2x - 4. The set of all solutions of an equation is called the solution set of the equation.
To solve an equation is to find all its solutions.
Equivalent EquationsEquations are equivalent if they have exactly the same solutions.
The following equations are equivalent since they have the same solution x = 0.
-3x + 2 = x + 2
-3x = x
x = 0
Properties of Equality
1 - Addition Property of EqualityIf we add the same number (or mathematical expression) to both sides of an equation, we do not change the solution set of the equation.
If A = B then A + C = B + C
The equation 2x + 3 = 5
and the equation 2x + 3 + (-3) = 5 + (-3) have the same solution x = 1.
2 - Multiplication Property of EqualityIf we multiply both sides of an equation by the same number (or mathematical expression), we do not change the solution set of the equation.
If A = B then C × A = C × B , with C not equal to zero.
The equation x / 2 = 4
and the equation 2 * (x / 2) = 2 * 4 have the same solution x = 8.
More References and linksSolve Equations, Systems of Equations and Inequalities.