# Equivalent Equations

## equivalent equations

[i¦kwiv·ə·lənt i′kwā·zhənz]## 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, 3*x* – 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, *x*^{2} - *x* + 1 = *x* – 1 and *x*^{2} - 2*x* + =0 are equivalent equations, since the polynomial - *x* + 1 has been added to both sides of the first equation. The equations 0.01 *x*^{2} – 0.37. *x* + 1 = 0 and *x*^{2} – 37*x* + 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, *x*^{2} - 1 = 0 and *x*^{4} – 1 = 0 are equivalent equations in the field of real numbers. The solution sets for both equations consist of two numbers: *x*, = 1 and *x*_{2} = – 1. The two equations, however, are not equivalent in the field of complex numbers, since the second equation now has two additional imaginary roots: *x*_{3} = *i* and *x*_{4} = – *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.

A. I. MARKUSHEVICH