# 4元1次方程式の解

${a}_{11}{x}_{1}+{a}_{12}{x}_{2}+{a}_{13}{x}_{3}+{a}_{14}{x}_{4}={b}_{1}$　　･･････(1)

${a}_{21}{x}_{1}+{a}_{22}{x}_{2}+{a}_{23}{x}_{3}+{a}_{24}{x}_{4}={b}_{2}$ 　　･･････(2)

${a}_{31}{x}_{1}+{a}_{32}{x}_{2}+{a}_{33}{x}_{3}+{a}_{34}{x}_{4}={b}_{3}$ 　　･･････(3)

${a}_{41}{x}_{1}+{a}_{42}{x}_{2}+{a}_{43}{x}_{3}+{a}_{44}{x}_{4}={b}_{4}$ 　　･･････(4)

${x}_{1}=\frac{{a}_{14}{}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){C}_{2}}{{a}_{14}{}^{2}\left({a}_{13}{a}_{24}-{a}_{14}{a}_{23}\right){C}_{1}}$

$\begin{array}{l}{C}_{1}={a}_{11}{a}_{22}{a}_{33}{a}_{44}-{a}_{11}{a}_{22}{a}_{34}{a}_{43}-{a}_{11}{a}_{23}{a}_{32}{a}_{44}+{a}_{11}{a}_{23}{a}_{34}{a}_{42}\\ \text{ }+{a}_{11}{a}_{24}{a}_{32}{a}_{43}-{a}_{11}{a}_{24}{a}_{33}{a}_{42}-{a}_{12}{a}_{21}{a}_{33}{a}_{44}+{a}_{12}{a}_{21}{a}_{34}{a}_{43}\\ \text{ }+{a}_{12}{a}_{23}{a}_{31}{a}_{44}-{a}_{12}{a}_{23}{a}_{34}{a}_{41}-{a}_{12}{a}_{24}{a}_{31}{a}_{43}+{a}_{12}{a}_{24}{a}_{33}{a}_{41}\\ \text{ }+{a}_{13}{a}_{21}{a}_{32}{a}_{44}-{a}_{13}{a}_{21}{a}_{34}{a}_{42}-{a}_{13}{a}_{22}{a}_{31}{a}_{44}+{a}_{13}{a}_{22}{a}_{34}{a}_{41}\\ \text{ }+{a}_{13}{a}_{24}{a}_{31}{a}_{42}-{a}_{13}{a}_{24}{a}_{32}{a}_{41}-{a}_{14}{a}_{21}{a}_{32}{a}_{43}+{a}_{14}{a}_{21}{a}_{33}{a}_{42}\\ \text{ }+{a}_{14}{a}_{22}{a}_{31}{a}_{43}-{a}_{14}{a}_{22}{a}_{33}{a}_{41}-{a}_{14}{a}_{23}{a}_{31}{a}_{42}+{a}_{14}{a}_{23}{a}_{32}{a}_{41}\end{array}$

${C}_{2}={a}_{22}{a}_{33}{a}_{44}{b}_{1}-{a}_{22}{a}_{34}{a}_{43}{b}_{1}-{a}_{23}{a}_{32}{a}_{44}{b}_{1}+{a}_{23}{a}_{34}{a}_{42}{b}_{1}\\ \text{ }+{a}_{24}{a}_{32}{a}_{43}{b}_{1}-{a}_{24}{a}_{33}{a}_{42}{b}_{1}-{a}_{12}{a}_{33}{a}_{44}{b}_{2}+{a}_{12}{a}_{34}{a}_{43}{b}_{2}\\ \text{ }+{a}_{13}{{a}_{32}a}_{44}{b}_{2}-{a}_{13}{{a}_{34}a}_{42}{b}_{2}-{a}_{14}{a}_{32}{a}_{43}{b}_{2}+{a}_{14}{a}_{33}{a}_{42}{b}_{2}\\ \text{ }+{a}_{12}{a}_{23}{\mathrm{a44}}_{}{b}_{3}-{a}_{12}{a}_{24}{a}_{43}{b}_{3}-{a}_{13}{a}_{22}{a}_{44}{b}_{3}+{a}_{13}{{a}_{24}{a}_{42}b}_{3}\\ \text{ }+{a}_{14}{a}_{22}{a}_{43}{b}_{3}-{a}_{14}{a}_{23}{a}_{42}{b}_{3}-{a}_{12}{a}_{23}{a}_{34}{b}_{4}+{a}_{12}{a}_{24}{a}_{33}{b}_{4}\\ \text{ }+{a}_{13}{a}_{22}{a}_{34}{b}_{4}-{a}_{13}{a}_{24}{a}_{32}{b}_{4}-{a}_{14}{a}_{22}{a}_{33}{b}_{4}+{a}_{14}{a}_{23}{a}_{32}{b}_{4}$

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

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