Equation - equation

Japanese: 方程式 - ほうていしき

An equation is an equation that contains variables (letters that can take on various values) and is not an identity (an equation that holds true when any number is substituted for the variables). For example, 3 x +1= x +5, 2 x - y =1. The variables in an equation are called unknowns, and the other numbers or letters, that is, constants, are called knowns. An equation is, so to speak, an equation that conditions the possible values of the variables in the equation. In logical terms, an equation is a propositional function. The possible values of the unknowns are solutions or roots, and the set of all solutions is called the solution set. Finding the solution set is also called solving an equation. Two equations that have the same solution set are said to be equivalent. The general method for solving equations is to successively replace them with equivalent equations (equivalent transformation).

An equation in which both sides are polynomials in unknowns is called a polynomial equation, and the number of unknowns is called the number of elements, and the highest degree of the unknowns is called the degree of the equation. For example, x2 ^-2x +3=0 is a one-dimensional quadratic equation, and 2x3 + y = ³ is a two-dimensional cubic equation. A one-dimensional n- th degree equation with complex coefficients has exactly n solutions within the range of complex numbers (Fundamental Theorem of Equations). However, multiple roots are counted as the number of overlaps. Expressing the solution to a polynomial equation as an algebraic expression for the coefficients (an expression obtained by applying the four arithmetic operations and the polynomial operation a finite number of times) is called an algebraic solution. General polynomial equations of degree five or higher cannot be solved algebraically (Abel's Theorem).

For example, in ( a - b ) x = a2 - b2 ^, when a = b , 0 x =0, and there are an infinite ^number of solutions. In such cases, the equation is said to be indeterminate. Also, when a = 1 and a is zero, there is no solution. In such cases, the equation is said to be impossible.

The solution to a two-dimensional equation with x and y , for example x - y +1=0, is that when x is any real number α, then y takes the value α+1. If this solution is expressed as a point (α, α+1) on the coordinate plane, the set of such points will form a straight line with a y-intercept of 1 and a slope of 1. In this case, this straight line is called the graph of the given equation. In general, the graph of a two-dimensional equation f ( x , y )=0 will be a plane curve. In particular, the ^graph ^of the two-dimensional quadratic equation ax2 + bxy + cy2 + ex + fy + g =0
The graph of a quadratic curve (or conic curve) is either a circle, an ellipse, a parabola, a hyperbola, or a straight line. The graph of an equation is used in the geometric solution of the equation.

[Yoshio Takeuchi]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

変数（いろいろな値をとりうる文字）を含んだ等式で恒等式（変数に任意の数を代入して成り立つ等式）でないものを方程式という。たとえば3x+1=x+5,2x-y=1などである。方程式の中の変数を未知数といい、それ以外の数字または文字、すなわち定数を既知数という。方程式はいわば式中の変数のとりうる値を条件づける等式である。論理的用語でいえば、方程式は命題関数である。未知数のとりうる値を解または根(こん)、すべての解の集合を解集合という。また解集合を求めることを方程式を解くという。同一解集合をもつ二つの方程式は同値であるという。方程式の一般的解法は順次同値な方程式に置き換え（同値変形）てなされる。

　方程式の両辺が未知数についての整式であるものを整方程式といい、未知数の個数を元(げん)の数、未知数の最高次数を方程式の次数という。たとえばx²-2x+3=0は一元二次であり、2x³+y=3は二元三次である。複素数を係数にもつ一元n次方程式は複素数の範囲内で、ちょうどn個の解をもつ（方程式論の基本定理）。ただし重根はその重なった数だけ数えるものとする。整方程式の解を係数の代数式（四則と開法を有限回施した式）で表すことを、代数的解法という。五次以上の一般的な整方程式を代数的に解くことはできない（アーベルの定理）。

　たとえば(a-b)x=a²-b²においてa=bのときは0x=0となり、解が無限個ある。このような場合に方程式は不定であるという。またax=1においてaがゼロのとき、解が存在しない。このようなとき方程式は不能であるという。

　xとyの二元方程式、たとえばx-y+1=0の解は、xが任意の実数αをとるとき、yはα+1の値をとる。いまこの解を座標平面上の点（α,α+1）で表すと、このような点の集合は、y切片1、傾き1の直線となる。このとき、この直線を与えられた方程式のグラフという。一般に二元方程式f(x,y)=0のグラフは平面曲線となる。とくに二元二次方程式
　　ax²+bxy+cy²+ex+fy+g=0
のグラフは二次曲線（または円錐(えんすい)曲線）になる。つまり円、楕円(だえん)、放物線、双曲線および二直線のいずれかである。方程式のグラフは方程式の幾何学的解法に用いられる。