% Licence : Released under the LaTeX Project Public License v1.3c % or later, see http://www.latex-project.org/lppl.txtf \newcommand{\EquaBaseL}[5][]{%type ax=d ou b=cx \useKVdefault[ClesEquation]% \setKV[ClesEquation]{#1}% \ifx\bla#2\bla%on teste si le paramètre #2 est vide: % si oui, on est dans le cas b=cx. Eh bien on échange :) % Mais attention si les deux paramètres a et c sont vides... \EquaBaseL[#1]{#4}{}{}{#3} \else % si non, on est dans le cas ax=d \xintifboolexpr{#2==0}{% \xintifboolexpr{#5==0}{% L'équation $0\useKV[ClesEquation]{Lettre}=0$ a une infinité de solutions.}{L'équation $0\useKV[ClesEquation]{Lettre}=\num{#5}$ n'a aucune solution.}% }{%\else \xintifboolexpr{#5==0}{L'équation $\num{#2}\useKV[ClesEquation]{Lettre}=0$ a une unique solution : $\useKV[ClesEquation]{Lettre}=0$.}{%\else \begin{align*}% \xintifboolexpr{#2==1}{\useKV[ClesEquation]{Lettre}}{\num{#2}\useKV[ClesEquation]{Lettre}}&=\num{#5}\\ \xintifboolexpr{#2==1}{}{% \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{#2<0}{(\num{#2})}{\num{#2}}}\phantom{\useKV[ClesEquation]{Lettre}}&\phantom{=}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{#2<0}{(\num{#2})}{\num{#2}}}\\} \useKV[ClesEquation]{Lettre}&=\frac{\num{#5}}{\num{#2}}%\\ \ifboolKV[ClesEquation]{Decimal}{% \\\useKV[ClesEquation]{Lettre}&=\num{\fpeval{#5/#2}}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{#5}{#2}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{\\\useKV[ClesEquation]{Lettre}&=\SSimplifie{#5}{#2}}{}%\\ }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{\useKV[ClesEquation]{Lettre}=\num{#5}}{\num{#2}\useKV[ClesEquation]{Lettre}=\num{#5}}$ a une unique solution : $\displaystyle\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\opdiv*{#5}{#2}{numequa}{resteequa}\opcmp{resteequa}{0}\ifopeq\opexport{numequa}{\numequa}\num{\numequa}\else\ifboolKV[ClesEquation]{Simplification}{\SSimplifie{#5}{#2}}{\frac{\num{#5}}{\num{#2}}}\fi$.% }{} } } \fi } \newcommand{\EquaDeuxL}[5][]{%type ax+b=d ou b=cx+d$ \useKVdefault[ClesEquation]% \setKV[ClesEquation]{#1}% \ifx\bla#2\bla%On échange en faisant attention à ne pas boucler : c doit être non vide \EquaDeuxL[#1]{#4}{#5}{#2}{#3} \else%cas ax+b=d \xintifboolexpr{#2==0}{% \xintifboolexpr{#3==#5}{%b=d L'équation $\num{#3}=\num{#5}$ a une infinité de solutions.}% {%b<>d L'équation $\num{#3}=\num{#5}$ n'a aucune solution.% }% }{%ELSE \xintifboolexpr{#3==0}{%ax+b=d \EquaBaseL[#1]{#2}{}{}{#5}% }{%ax+b=d$ Ici \begin{align*} \xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\num{#5}\\ \phantom{\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}}\mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{-\num{#3}}{+\num{\fpeval{0-#3}}}}&\phantom{\mathrel{=}}\mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{-\num{#3}}{+\num{\fpeval{0-#3}}}}\\ \xdef\Coeffa{#2}\xdef\Coeffb{\fpeval{#5-#3}}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\num{\Coeffb}%\\ \xintifboolexpr{\Coeffa==1}{}{\\} \xintifboolexpr{\Coeffa==1}{% }{%\ifnum\cmtd>1 \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{#2<0}{(\num{#2})}{\num{#2}}}\phantom{\useKV[ClesEquation]{Lettre}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\phantom{=}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{#2<0}{(\num{#2})}{\num{#2}}}\\ \useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\frac{\num{\Coeffb}}{\num{\Coeffa}}%\\ } \ifboolKV[ClesEquation]{Decimal}{% \\\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\num{\fpeval{\Coeffb/\Coeffa}}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{\Coeffb}{\Coeffa}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{% \\\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\SSimplifie{\Coeffb}{\Coeffa}% }{}%\\ }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\num{#5}$ a une unique solution : \opdiv*{\Coeffb}{\Coeffa}{solution}{resteequa}\opcmp{resteequa}{0}$\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\displaystyle\ifopeq\opexport{solution}{\solution}\num{\solution}\else\ifboolKV[ClesEquation]{Entier}{\SSimplifie{\Coeffb}{\Coeffa}}{\frac{\num{\Coeffb}}{\num{\Coeffa}}}\fi$. }{} } } \fi } \newcommand{\EquaTroisL}[5][]{%ax+b=cx ou ax=cx+d \useKVdefault[ClesEquation]% \setKV[ClesEquation]{#1}% \ifx\bla#3\bla%on inverse en faisant attention à la boucle #3<->#5 \ifx\bla#5\bla% %% paramètre oublié \else \EquaTroisL[#1]{#4}{#5}{#2}{}% \fi \else \xintifboolexpr{#2==0}{%b=cx \EquaBaseL[#1]{#4}{}{}{#3} }{% \xintifboolexpr{#4==0}{%ax+b=0 \EquaDeuxL[#1]{#2}{#3}{}{0} }{%ax+b=cx \xintifboolexpr{#2==#4}{% \xintifboolexpr{#3==0}{%ax=ax L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}$ a une infinité de solutions.}% {%ax+b=ax L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}$ n'a aucune solution.% }% }{%% Cas délicat \xintifboolexpr{#2>#4}{%ax+b=cx avec a>c \begin{align*} \xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#4>0}{{}-{}\num{#4}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#4}}\useKV[ClesEquation]{Lettre}}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\phantom{{}={}}\mathcolor{Cdecomp}{\xintifboolexpr{#4>0}{-\num{#4}\useKV[ClesEquation]{Lettre}}{+\num{\fpeval{0-#4}}\useKV[ClesEquation]{Lettre}}}\\ \xdef\Coeffa{\fpeval{#2-#4}}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=0\\ \phantom{\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}}\mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{{}-{}\num{#3}}{{}+{}\num{\fpeval{0-#3}}}}&\phantom{\mathrel{=}}\mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{{}-{}\num{#3}}{{}+{}\num{\fpeval{0-#3}}}}\\ \xdef\Coeffb{\fpeval{0-#3}}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\num{\Coeffb}%\\ \xintifboolexpr{\Coeffa==1}{}{\\} \xintifboolexpr{\Coeffa==1}{% }{%\ifnum\cmtd>1 \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\phantom{\useKV[ClesEquation]{Lettre}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\phantom{=}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\\ \useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\frac{\num{\Coeffb}}{\num{\Coeffa}}%\\ } \ifboolKV[ClesEquation]{Decimal}{% \\\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\num{\fpeval{\Coeffb/\Coeffa}}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{\Coeffb}{\Coeffa}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{\\% \useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\SSimplifie{\Coeffb}{\Coeffa}%\\ }{} }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}$ a une unique solution : \opdiv*{\Coeffb}{\Coeffa}{solution}{resteequa}\opcmp{resteequa}{0}$\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\displaystyle\ifopeq\opexport{solution}{\solution}\num{\solution}\else\ifboolKV[ClesEquation]{Entier}{\SSimplifie{\Coeffb}{\Coeffa}}{\frac{\num{\Coeffb}}{\num{\Coeffa}}}\fi$.}{} }{%ax+b=cx+d avec a0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#2>0}{{}-{}\num{#2}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#2}}\useKV[ClesEquation]{Lettre}}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\phantom{{}={}}\mathcolor{Cdecomp}{\xintifboolexpr{#2>0}{{}-{}\num{#2}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#2}}\useKV[ClesEquation]{Lettre}}}\\ \xdef\Coeffb{#3}\xdef\Coeffa{\fpeval{#4-#2}}\xintifboolexpr{#3>0}{\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}%\\ \xintifboolexpr{\Coeffa==1}{}{\\} \xintifboolexpr{\Coeffa==1}{% }{%\ifnum\cmtd>1 \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}&\phantom{=}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\\ \frac{\num{\Coeffb}}{\num{\Coeffa}}&=\phantom{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}%\\ } \ifboolKV[ClesEquation]{Decimal}{% \\\num{\fpeval{\Coeffb/\Coeffa}}&=\phantom{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{\Coeffb}{\Coeffa}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{\\% \SSimplifie{\Coeffb}{\Coeffa}&=\phantom{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}%\\ }{} }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}$ a une unique solution : \opdiv*{\Coeffb}{\Coeffa}{solution}{resteequa}\opcmp{resteequa}{0}$\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\displaystyle\ifopeq\opexport{solution}{\solution}\num{\solution}\else\ifboolKV[ClesEquation]{Entier}{\SSimplifie{\Coeffb}{\Coeffa}}{\frac{\num{\Coeffb}}{\num{\Coeffa}}}\fi$.}{}% }% }% }% }% \fi }%\\ % \\ \newcommand{\ResolEquationL}[5][]{% \useKVdefault[ClesEquation]% \setKV[ClesEquation]{#1}% \xintifboolexpr{#2==0}{% \xintifboolexpr{#4==0}{% \xintifboolexpr{#3==#5}{%b=d L'équation $\num{#3}=\num{#5}$ a une infinité de solutions.}% {%b<>d L'équation $\num{#3}=\num{#5}$ n'a aucune solution.% }% }% {%0x+b=cx+d$ \EquaDeuxL[#1]{#4}{#5}{}{#3}% }% }{% \xintifboolexpr{#4==0}{%ax+b=0x+d \EquaDeuxL[#1]{#2}{#3}{}{#5}% } {%ax+b=cx+d$ \xintifboolexpr{#3==0}{% \xintifboolexpr{#5==0}{%ax=cx \EquaTroisL[#1]{#2}{0}{#4}{}% }% {%ax=cx+d \EquaTroisL[#1]{#4}{#5}{#2}{}% }% }% {\xintifboolexpr{#5==0}{%ax+b=cx \EquaTroisL[#1]{#2}{#3}{#4}{}% }% {%ax+b=cx+d -- ici \xintifboolexpr{#2==#4}{% \xintifboolexpr{#3==#5}{%b=d L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}$ a une infinité de solutions.}% {%b<>d L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}$ n'a aucune solution.% }% }{ %% Cas délicat \xintifboolexpr{#2>#4}{%ax+b=cx+d avec a>c \begin{align*} \xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#4>0}{{}-{}\num{#4}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#4}}\useKV[ClesEquation]{Lettre}}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\mathcolor{Cdecomp}{\xintifboolexpr{#4>0}{{}-{}\num{#4}\useKV[ClesEquation]{Lettre}}{\phantom{{}={}}+{}\num{\fpeval{0-#4}}\useKV[ClesEquation]{Lettre}}}\\ \xdef\Coeffa{\fpeval{#2-#4}}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}}\xintifboolexpr{#5>0}{\phantom{{}+{}}\num{#5}}{-\num{\fpeval{0-#5}}}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{{}-{}\num{#3}}{{}+{}\num{\fpeval{0-#3}}}}&\phantom{{}={}\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}}\mathcolor{Cdecomp}{\xintifboolexpr{#3>0}{{}-{}\num{#3}}{{}+{}\num{\fpeval{0-#3}}}}\\ \xdef\Coeffb{\fpeval{#5-#3}}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}}\xintifboolexpr{\Coeffb>0}{\phantom{{}+{}}\num{\Coeffb}}{{}-{}\num{\fpeval{0-\Coeffb}}}%\\ \xintifboolexpr{\Coeffa==1}{}{\\} \xintifboolexpr{\Coeffa==1}{}{%\ifnum\cmtd>1 \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\phantom{\useKV[ClesEquation]{Lettre}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\phantom{{}={}}\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\\ \phantom{\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}}\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{\Coeffb>0}{{}+{}}{}}\frac{\num{\Coeffb}}{\num{\Coeffa}}%\\ } \ifboolKV[ClesEquation]{Decimal}{% \\\useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{\Coeffb>0}{{}+{}}{}}\num{\fpeval{\Coeffb/\Coeffa}}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{\Coeffb}{\Coeffa}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{\\% \useKV[ClesEquation]{Lettre}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&=\phantom{\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{\Coeffb>0}{{}+{}}{}}\SSimplifie{\Coeffb}{\Coeffa}%\\ }{} }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}$ a une unique solution : \opdiv*{\Coeffb}{\Coeffa}{solution}{resteequa}\opcmp{resteequa}{0}$\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\displaystyle\ifopeq\opexport{solution}{\solution}\num{\solution}\else\ifboolKV[ClesEquation]{Entier}{\SSimplifie{\Coeffb}{\Coeffa}}{\frac{\num{\Coeffb}}{\num{\Coeffa}}}\fi$.% }{} }{%ax+b=cx+d avec a0}{+\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#2>0}{{}-{}\num{#2}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#2}}\useKV[ClesEquation]{Lettre}}}\phantom{\xintifboolexpr{#3>0}{{}+{}\num{#3}}{{}-{}\num{\fpeval{0-#3}}}}&\xintifboolexpr{#4<0}{\phantom{={}}}{}\mathcolor{Cdecomp}{\xintifboolexpr{#2>0}{{}-{}\num{#2}\useKV[ClesEquation]{Lettre}}{{}+{}\num{\fpeval{0-#2}}\useKV[ClesEquation]{Lettre}}}\\ \xdef\Coeffa{\fpeval{#4-#2}}\xintifboolexpr{#3>0}{\num{#3}}{-\num{\fpeval{0-#3}}}&=\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}\\ \mathcolor{Cdecomp}{\xintifboolexpr{#5>0}{{}-{}\num{#5}}{{}+{}\num{\fpeval{0-#5}}}}&\phantom{{}={}\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}}\mathcolor{Cdecomp}{\xintifboolexpr{#5>0}{-\num{#5}}{+\num{\fpeval{0-#5}}}}\\ \xdef\Coeffb{\fpeval{#3-#5}}\num{\Coeffb}&=\xintifboolexpr{\Coeffa==1}{}{\num{\Coeffa}}\useKV[ClesEquation]{Lettre}%\\ \xintifboolexpr{\Coeffa==1}{}{\\} \xintifboolexpr{\Coeffa==1}{}{%\ifnum\cmtd>1 \mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}&\xintifboolexpr{\Coeffa<0}{\phantom{{}={}}}{\phantom{=}}\mathcolor{Cdecomp}{\mathrel{\div}\xintifboolexpr{\Coeffa<0}{(\num{\Coeffa})}{\num{\Coeffa}}}\\ \frac{\num{\Coeffb}}{\num{\Coeffa}}&=\useKV[ClesEquation]{Lettre}%\\ } \ifboolKV[ClesEquation]{Decimal}{% \\\num{\fpeval{\Coeffb/\Coeffa}}&=\useKV[ClesEquation]{Lettre}% }{}% % %%% \ifboolKV[ClesEquation]{Entier}{% \SSimpliTest{\Coeffb}{\Coeffa}% \ifboolKV[ClesEquation]{Simplification}{% \ifthenelse{\boolean{Simplification}}{\\\SSimplifie{\Coeffb}{\Coeffa}&=\useKV[ClesEquation]{Lettre}}{}%\\ }{} }{} \end{align*} \ifboolKV[ClesEquation]{Solution}{L'équation $\xintifboolexpr{#2==1}{}{\num{#2}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#3>0}{+\num{#3}}{-\num{\fpeval{0-#3}}}=\xintifboolexpr{#4==1}{}{\num{#4}}\useKV[ClesEquation]{Lettre}\xintifboolexpr{#5>0}{+\num{#5}}{-\num{\fpeval{0-#5}}}$ a une unique solution : \opdiv*{\Coeffb}{\Coeffa}{solution}{resteequa}\opcmp{resteequa}{0}$\ifboolKV[ClesEquation]{LettreSol}{\useKV[ClesEquation]{Lettre}=}{}\displaystyle\ifopeq\opexport{solution}{\solution}\num{\solution}\else\ifboolKV[ClesEquation]{Entier}{\SSimplifie{\Coeffb}{\Coeffa}}{\frac{\num{\Coeffb}}{\num{\Coeffa}}}\fi$.% }{}% }% }% }% }% }% }% }%