Question Number 59942 by naka3546 last updated on 16/May/19
$${f}\left(\mathrm{2}{x}−\mathrm{5}\right)\:+\:{f}\left(\mathrm{3}{x}+\mathrm{5}\right)\:\:=\:\:\mathrm{7}{x}−\mathrm{16} \\ $$$${f}\left({x}\right)\:\:=\:\:? \\ $$
Answered by mr W last updated on 16/May/19
$${let}\:{f}\left({x}\right)={ax}+{b} \\ $$$${f}\left(\mathrm{2}{x}−\mathrm{5}\right)+{f}\left(\mathrm{3}{x}+\mathrm{5}\right)={a}\left(\mathrm{2}{x}−\mathrm{5}\right)+{b}+{a}\left(\mathrm{3}{x}+\mathrm{5}\right)+{b} \\ $$$$=\mathrm{5}{ax}+\mathrm{2}{b} \\ $$$$=\mathrm{7}{x}−\mathrm{16} \\ $$$$\Rightarrow\mathrm{5}{a}=\mathrm{7}\Rightarrow{a}=\frac{\mathrm{7}}{\mathrm{5}} \\ $$$$\Rightarrow\mathrm{2}{b}=−\mathrm{16}\Rightarrow{b}=−\mathrm{8} \\ $$$$\Rightarrow{f}\left({x}\right)=\frac{\mathrm{7}}{\mathrm{5}}{x}−\mathrm{8} \\ $$