Solve {l}{5^-1x+6^1/8y=729}{6^17x+5^18y=629}

$$\left. \begin{array} { l } { 5 ^ { - 1 } x + 6 ^ { 1 / 8 } y = 729 } \\ { 6 ^ { 17 } x + 5 ^ { 18 } y = 629 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{{(82089011515213367907186323883068205046425814529212416 \sqrt{6} + 338813178901720135627329000271856784820556640625)} {(2780914306640625 - 629 \sqrt[8]{6})} {(16926659444736 \sqrt[8]{6} + 762939453125)} {(286511799958070431838109696 \sqrt[4]{6} + 582076609134674072265625)}}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711} = -136\frac{9.43412243425362 \times 10^{105}}{4.04316348691542 \times 10^{106}} \approx -136.233335171$$

$$y = \frac{61697673676062091 {(82089011515213367907186323883068205046425814529212416 \sqrt{6} + 338813178901720135627329000271856784820556640625)} {(16926659444736 \sqrt[8]{6} + 762939453125)} {(286511799958070431838109696 \sqrt[4]{6} + 582076609134674072265625)}}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555} = 604\frac{1.0059007409351448 \times 10^{107}}{2.0215817434577104 \times 10^{107}} \approx 604.497581037$$

Steps Using Substitution

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$\frac{1}{5}x+\sqrt[8]{6}y=729,16926659444736x+3814697265625y=629$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$\frac{1}{5}x+\sqrt[8]{6}y=729$$

Subtract \sqrt[8]{6}y from both sides of the equation.

$$\frac{1}{5}x=\left(-\sqrt[8]{6}\right)y+729$$

Multiply both sides by 5.

$$x=5\left(\left(-\sqrt[8]{6}\right)y+729\right)$$

Multiply 5 times -\sqrt[8]{6}y+729.

$$x=\left(-5\sqrt[8]{6}\right)y+3645$$

Substitute -5\sqrt[8]{6}y+3645 for x in the other equation, 16926659444736x+3814697265625y=629.

$$16926659444736\left(\left(-5\sqrt[8]{6}\right)y+3645\right)[phone]y=629$$

Multiply 16926659444736 times -5\sqrt[8]{6}y+3645.

$$\left(-84633297223680\sqrt[8]{6}\right)y+61697673676062720+3814697265625y=629$$

Add -84633297223680\sqrt[8]{6}y to 3814697265625y.

$$\left(3814697265625-84633297223680\sqrt[8]{6}\right)y+61697673676062720=629$$

Subtract 61697673676062720 from both sides of the equation.

$$\left(3814697265625-84633297223680\sqrt[8]{6}\right)y=-61697673676062091$$

Divide both sides by -84633297223680\sqrt[8]{6}[phone].

$$y=\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}$$

Substitute \frac{61697673676062091\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555} for y in x=\left(-5\sqrt[8]{6}\right)y+3645. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-5\sqrt[8]{6}\right)\times \frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}+3645$$

Multiply -5\sqrt[8]{6} times \frac{61697673676062091\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}.

$$x=-\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\sqrt[8]{6}\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}+3645$$

The system is now solved.

$$x=-\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\sqrt[8]{6}\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}+3645,y=\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$\frac{1}{5}x+\sqrt[8]{6}y=729,16926659444736x+3814697265625y=629$$

Write the equations in matrix form.

$$\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}729\\629\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right))\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right))\left(\begin{matrix}729\\629\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right))\left(\begin{matrix}729\\629\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&\sqrt[8]{6}\\16926659444736&3814697265625\end{matrix}\right))\left(\begin{matrix}729\\629\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3814697265625}{\frac{1}{5}\times 3814697265625-\sqrt[8]{6}\times 16926659444736}&-\frac{\sqrt[8]{6}}{\frac{1}{5}\times 3814697265625-\sqrt[8]{6}\times 16926659444736}\\-\frac{16926659444736}{\frac{1}{5}\times 3814697265625-\sqrt[8]{6}\times 16926659444736}&\frac{\frac{1}{5}}{\frac{1}{5}\times 3814697265625-\sqrt[8]{6}\times 16926659444736}\end{matrix}\right)\left(\begin{matrix}729\\629\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3814697265625\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}&\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\sqrt[8]{6}\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}\\\frac{16926659444736\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}&-\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}\end{matrix}\right)\left(\begin{matrix}729\\629\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{3814697265625\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}\right)\times 729+\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\sqrt[8]{6}\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}\times 629\\\frac{16926659444736\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}\times 729+\left(-\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}\right)\times 629\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=UndefinedLatexOperatorName(\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(629\sqrt[8]{6}-2780914306640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711},\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555})$$

Extract the matrix elements x and y.

$$x=\frac{\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(629\sqrt[8]{6}-2780914306640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711},y=\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}$$

Steps Using Elimination

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$\frac{1}{5}x+\sqrt[8]{6}y=729,16926659444736x+3814697265625y=629$$

To make \frac{x}{5} and 16926659444736x equal, multiply all terms on each side of the first equation by 16926659444736 and all terms on each side of the second by \frac{1}{5}.

$$16926659444736\times \frac{1}{5}x+16926659444736\sqrt[8]{6}y=16926659444736\times 729,\frac{1}{5}\times 16926659444736x+\frac{1}{5}\times 3814697265625y=\frac{1}{5}\times 629$$

Simplify.

$$\frac{16926659444736}{5}x+16926659444736\sqrt[8]{6}y=12339534735212544,\frac{16926659444736}{5}x+762939453125y=\frac{629}{5}$$

Subtract \frac{16926659444736}{5}x+762939453125y=\frac{629}{5} from \frac{16926659444736}{5}x+16926659444736\sqrt[8]{6}y=12339534735212544 by subtracting like terms on each side of the equal sign.

$$\frac{16926659444736}{5}x-\frac{16926659444736}{5}x+16926659444736\sqrt[8]{6}y-762939453125y=12339534735212544-\frac{629}{5}$$

Add \frac{16926659444736x}{5} to -\frac{16926659444736x}{5}. Terms \frac{16926659444736x}{5} and -\frac{16926659444736x}{5} cancel out, leaving an equation with only one variable that can be solved.

$$16926659444736\sqrt[8]{6}y-762939453125y=12339534735212544-\frac{629}{5}$$

Add 16926659444736\sqrt[8]{6}y to -762939453125y.

$$\left(16926659444736\sqrt[8]{6}-762939453125\right)y=12339534735212544-\frac{629}{5}$$

Add 12339534735212544 to -\frac{629}{5}.

$$\left(16926659444736\sqrt[8]{6}-762939453125\right)y=\frac{61697673676062091}{5}$$

Divide both sides by 16926659444736\sqrt[8]{6}-762939453125.

$$y=\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}$$

Substitute \frac{61697673676062091\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555} for y in 16926659444736x+3814697265625y=629. Because the resulting equation contains only one variable, you can solve for x directly.

$$16926659444736x+3814697265625\times \frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}=629$$

Multiply 3814697265625 times \frac{61697673676062091\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}.

$$16926659444736x+\frac{47071589413499520111083984375\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}=629$$

Subtract \frac{47071589413499520111083984375\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711} from both sides of the equation.

$$16926659444736x=-\frac{47071589413499520111083984375\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{40431634869154203128232179250585295273911708459631231043986648578686056486646640972710718280400390090951711}+629$$

Divide both sides by 16926659444736.

$$x=-\frac{47071589413499520111083984375\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{684372514224086379936483053438999314704116964141990687134331170622699688494727762885783153616877791688666485053049143296}+\frac{629}{16926659444736}$$

The system is now solved.

$$x=-\frac{47071589413499520111083984375\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{684372514224086379936483053438999314704116964141990687134331170622699688494727762885783153616877791688666485053049143296}+\frac{629}{16926659444736},y=\frac{61697673676062091\left(82089011515213367907186323883068205046425814529212416\sqrt{6}+338813178901720135627329000271856784820556640625\right)\left(16926659444736\sqrt[8]{6}[phone]\right)\left(286511799958070431838109696\sqrt[4]{6}+582076609134674072265625\right)}{202158174345771015641160896252926476369558542298156155219933242893430282433233204863553591402001950454758555}$$