Solve {l}{65^2=a^2+b^2}{a/b=16/9}

$$\left. \begin{array} { l } { 65 ^ { 2 } = a ^ { 2 } + b ^ { 2 } } \\ { \frac { a } { b } = \frac { 16 } { 9 } } \end{array} \right.$$

Solve for a, b

$$a=-\frac{1040\sqrt{337}}{337}\approx -56.652409913\text{, }b=-\frac{585\sqrt{337}}{337}\approx -31.866980576$$

$$a=\frac{1040\sqrt{337}}{337}\approx 56.652409913\text{, }b=\frac{585\sqrt{337}}{337}\approx 31.866980576$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Calculate 65 to the power of 2 and get 4225.

$$4225=a^{2}+b^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$a^{2}+b^{2}=4225$$

Consider the second equation. Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9b, the least common multiple of b,9.

$$9a=16b$$

Divide both sides by 9.

$$a=\frac{16}{9}b$$

Substitute \frac{16}{9}b for a in the other equation, b^{2}+a^{2}=4225.

$$b^{2}+\left(\frac{16}{9}b\right)^{2}=4225$$

Square \frac{16}{9}b.

$$b^{2}+\frac{256}{81}b^{2}=4225$$

Add b^{2} to \frac{256}{81}b^{2}.

$$\frac{337}{81}b^{2}=4225$$

Subtract 4225 from both sides of the equation.

$$\frac{337}{81}b^{2}-4225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times \left(\frac{16}{9}\right)^{2} for a, 1\times 0\times \frac{16}{9}\times 2 for b, and -4225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{0±\sqrt{0^{2}-4\times \frac{337}{81}\left(-4225\right)}}{2\times \frac{337}{81}}$$

Square 1\times 0\times \frac{16}{9}\times 2.

$$b=\frac{0±\sqrt{-4\times \frac{337}{81}\left(-4225\right)}}{2\times \frac{337}{81}}$$

Multiply -4 times 1+1\times \left(\frac{16}{9}\right)^{2}.

$$b=\frac{0±\sqrt{-\frac{1348}{81}\left(-4225\right)}}{2\times \frac{337}{81}}$$

Multiply -\frac{1348}{81} times -4225.

$$b=\frac{0±\sqrt{\frac{5695300}{81}}}{2\times \frac{337}{81}}$$

Take the square root of \frac{5695300}{81}.

$$b=\frac{0±\frac{130\sqrt{337}}{9}}{2\times \frac{337}{81}}$$

Multiply 2 times 1+1\times \left(\frac{16}{9}\right)^{2}.

$$b=\frac{0±\frac{130\sqrt{337}}{9}}{\frac{674}{81}}$$

Now solve the equation b=\frac{0±\frac{130\sqrt{337}}{9}}{\frac{674}{81}} when ± is plus.

$$b=\frac{585\sqrt{337}}{337}$$

Now solve the equation b=\frac{0±\frac{130\sqrt{337}}{9}}{\frac{674}{81}} when ± is minus.

$$b=-\frac{585\sqrt{337}}{337}$$

There are two solutions for b: \frac{585\sqrt{337}}{337} and -\frac{585\sqrt{337}}{337}. Substitute \frac{585\sqrt{337}}{337} for b in the equation a=\frac{16}{9}b to find the corresponding solution for a that satisfies both equations.

$$a=\frac{16}{9}\times \frac{585\sqrt{337}}{337}$$

Multiply \frac{16}{9} times \frac{585\sqrt{337}}{337}.

$$a=\frac{16\times \frac{585\sqrt{337}}{337}}{9}$$

Now substitute -\frac{585\sqrt{337}}{337} for b in the equation a=\frac{16}{9}b and solve to find the corresponding solution for a that satisfies both equations.

$$a=\frac{16}{9}\left(-\frac{585\sqrt{337}}{337}\right)$$

Multiply \frac{16}{9} times -\frac{585\sqrt{337}}{337}.

$$a=\frac{16\left(-\frac{585\sqrt{337}}{337}\right)}{9}$$

The system is now solved.

$$a=\frac{16\times \frac{585\sqrt{337}}{337}}{9},b=\frac{585\sqrt{337}}{337}\text{ or }a=\frac{16\left(-\frac{585\sqrt{337}}{337}\right)}{9},b=-\frac{585\sqrt{337}}{337}$$