Equivalent Equations

equations that have the same solution sets; in the case of multiple roots, the multiplicities of the respective roots must be equal. Thus, of the three equations = 2, 3x – 7 = 5, and (x – 4)₂ = 0, the first and second are equivalent but the first and third are not, since the multiplicity of the root x = 4, is equal to 1 for the first equation and 2 for the third equation.

If we add the same polynomial in x to both sides of a given equation or multiply both sides by a number other than zero, we obtain an equation equivalent to the given equation. For example, x² - x + 1 = x – 1 and x² - 2x + =0 are equivalent equations, since the polynomial - x + 1 has been added to both sides of the first equation. The equations 0.01 x² – 0.37. x + 1 = 0 and x² – 37x + 100 = 0 are also equivalent; here, both sides of the first equation have been multiplied by 100. If, however, we multiply or divide both sides of an equation by a polynomial of degree at least 1, then the resultant equation will in general not be equivalent to the original equation. For example, x – 1 = 0 and (x – 1) (x + 1) = 0 are not equivalent, since the root x = – 1 of the second equation is not a root of the first equation.

The concept of equivalent equations acquires a more precise meaning when the field in which the roots of the equations lie is indicated. For example, x² - 1 = 0 and x⁴ – 1 = 0 are equivalent equations in the field of real numbers. The solution sets for both equations consist of two numbers: x, = 1 and x₂ = – 1. The two equations, however, are not equivalent in the field of complex numbers, since the second equation now has two additional imaginary roots: x₃ = i and x₄ = – i.

The concept of equivalent equations can also be applied to a system of equations. For example, if P(x, y) and Q(x, y) are two polynomials in the variables x and y and if a, b, c, and d are real or complex numbers, then the two systems P(x, y) = 0, Q(x, y) = 0 and a P(x, y) + bQ(x, y) = 0, cP(x, y) + dQ(x, y) = 0 are equivalent when the value of the determinant ad – bc ≠ 0.