# 多元1次方程式の解

${x}_{1}$ のみ求めている．  解を求める計算をすることにより，行列式の定義の理解を深める．

## ■1元1次方程式

 ${a}_{11}{x}_{1}$ $={b}_{1}$ ${x}_{1}$ $=\frac{{b}_{1}}{{a}_{11}}$

## ■2元1次方程式

$\left\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}={b}_{2}\end{array}$

${x}_{1}=\frac{\left|\begin{array}{cc}{b}_{1}& {a}_{12}\\ {b}_{2}& {a}_{22}\end{array}\right|}{\left|\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right|}$

## ■3元1次方程式

$\left\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+{a}_{13}{x}_{3}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+{a}_{23}{x}_{3}={b}_{2}\\ {a}_{31}{x}_{1}+{a}_{32}{x}_{2}+{a}_{33}{x}_{3}={b}_{3}\end{array}$

${x}_{1}=\frac{{a}_{13}\left|\begin{array}{ccc}{b}_{1}& {a}_{12}& {a}_{13}\\ {b}_{2}& {a}_{22}& {a}_{23}\\ {b}_{3}& {a}_{32}& {a}_{33}\end{array}\right|}{{a}_{13}\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|}=\frac{\left|\begin{array}{ccc}{b}_{1}& {a}_{12}& {a}_{13}\\ {b}_{2}& {a}_{22}& {a}_{23}\\ {b}_{3}& {a}_{32}& {a}_{33}\end{array}\right|}{\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|}$

## ■4元1次方程式

$\left\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+{a}_{13}{x}_{3}+{a}_{14}{x}_{4}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+{a}_{23}{x}_{3}+{a}_{24}{x}_{4}={b}_{2}\\ {a}_{31}{x}_{1}+{a}_{32}{x}_{2}+{a}_{33}{x}_{3}+{a}_{34}{x}_{4}={b}_{3}\\ {a}_{41}{x}_{1}+{a}_{42}{x}_{2}+{a}_{43}{x}_{3}+{a}_{44}{x}_{4}={b}_{4}\end{array}$

${x}_{1}=\frac{{{a}_{14}}^{2}\left|\begin{array}{cc}{a}_{13}& {a}_{23}\\ {a}_{14}& {a}_{24}\end{array}\right|\left|\begin{array}{cccc}{b}_{1}& {a}_{12}& {a}_{13}& {a}_{14}\\ {b}_{2}& {a}_{22}& {a}_{23}& {a}_{24}\\ {b}_{3}& {a}_{32}& {a}_{33}& {a}_{34}\\ {b}_{4}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right|}{{{a}_{14}}^{2}\left|\begin{array}{cc}{a}_{13}& {a}_{23}\\ {a}_{14}& {a}_{24}\end{array}\right|\left|\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ {a}_{21}& {a}_{22}& {a}_{23}& {a}_{24}\\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right|}=\frac{\left|\begin{array}{cccc}{b}_{1}& {a}_{12}& {a}_{13}& {a}_{14}\\ {b}_{2}& {a}_{22}& {a}_{23}& {a}_{24}\\ {b}_{3}& {a}_{32}& {a}_{33}& {a}_{34}\\ {b}_{4}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right|}{\left|\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ {a}_{21}& {a}_{22}& {a}_{23}& {a}_{24}\\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right|}$

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