Category: Algebra

Question Number 216076 by efronzo1 last updated on 27/Jan/25 $$\frac{\lfloor \frac{\mathrm{x}}{3}\rfloor }{\lfloor \frac{\mathrm{x}}{4}\rfloor }=\frac{21}{16};\mathrm{x}=?$$ Answered by mr W last updated on 28/Jan/25 $$\lfloor \frac{x}{3}\rfloor =21k$$$$\lfloor \frac{x}{4}\rfloor =16k\ne 0\Rightarrow k\ne 0$$$$\frac{{x}}{\mathrm{3}}=\mathrm{21}{k}+{a}\:{with}\:\mathrm{0}\leqslant{a}<\mathrm{1}…

Question Number 216046 by MATHEMATICSAM last updated on 26/Jan/25 $$x=bz+cy,y=cx+az\mathrm{and}z=bx+ay$$$$\mathrm{then}\mathrm{prove}\mathrm{that}{a}^2+b^2+c^2+2abc=1.$$ Answered by A5T last updated on 26/Jan/25 $$\mathrm{x}=\mathrm{bz}+\mathrm{cy}…\left(\mathrm{i}\right);\:\mathrm{y}=\mathrm{cx}+\mathrm{az}…\left(\mathrm{ii}\right);\:\mathrm{z}=\mathrm{bx}+\mathrm{ay}…\left(\mathrm{iii}\right)…

Question Number 216010 by MATHEMATICSAM last updated on 25/Jan/25 $$\mathrm{Solve}\mathrm{for}x\mathrm{and}y$$$${ax}^2+bxy+{cy}^2={bx}^2+cxy+{ay}^2=d.$$ Answered by mr W last updated on…

Question Number 215992 by MATHEMATICSAM last updated on 24/Jan/25 $${(\sqrt{3}+\sqrt{2})}^x+{(\sqrt{3}-\sqrt{2})}^x=10.\mathrm{Solve}$$$$\mathrm{for}x.$$ Answered by Rasheed.Sindhi last updated on 24/Jan/25 $$\left(\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\right)^{{x}} \:+\:\left(\sqrt{\mathrm{3}}\:−\:\sqrt{\mathrm{2}}\right)^{{x}}…

Question Number 215973 by MATHEMATICSAM last updated on 23/Jan/25 $$(a+b)\propto c\mathrm{and}(b+c)\propto a.\mathrm{Prove}\mathrm{that}$$$$(c+a)\propto b.$$ Answered by Rasheed.Sindhi last updated on 24/Jan/25 $$(a+b)\propto c\mathrm{and}(b+c)\propto a.\mathrm{Prove}\mathrm{that}$$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \…