Solve {l}{x^2+y^2=1069}{100y+301x=104}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 1069 } \\ { 100 y + 301 x = 104 } \end{array} \right.$$

Solve for x, y

$$x=\frac{100\sqrt{107531653}+31304}{100601}\approx 10.618967016\text{, }y=\frac{10400-301\sqrt{107531653}}{100601}\approx -30.923090717$$

$$x=\frac{31304-100\sqrt{107531653}}{100601}\approx -9.996627278\text{, }y=\frac{301\sqrt{107531653}+10400}{100601}\approx 31.129848106$$

Steps Using Substitution and Quadratic Formula

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.

$$100y+301x=104,x^{2}+y^{2}=1069$$

Solve 100y+301x=104 for y by isolating y on the left hand side of the equal sign.

$$100y+301x=104$$

Subtract 301x from both sides of the equation.

$$100y=-301x+104$$

Divide both sides by 100.

$$y=-\frac{301}{100}x+\frac{26}{25}$$

Substitute -\frac{301}{100}x+\frac{26}{25} for y in the other equation, x^{2}+y^{2}=1069.

$$x^{2}+\left(-\frac{301}{100}x+\frac{26}{25}\right)^{2}=1069$$

Square -\frac{301}{100}x+\frac{26}{25}.

$$x^{2}+\frac{90601}{10000}x^{2}-\frac{3913}{625}x+\frac{676}{625}=1069$$

Add x^{2} to \frac{90601}{10000}x^{2}.

$$\frac{100601}{10000}x^{2}-\frac{3913}{625}x+\frac{676}{625}=1069$$

Subtract 1069 from both sides of the equation.

$$\frac{100601}{10000}x^{2}-\frac{3913}{625}x-\frac{667449}{625}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-\frac{301}{100}\right)^{2} for a, 1\times \frac{26}{25}\left(-\frac{301}{100}\right)\times 2 for b, and -\frac{667449}{625} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\sqrt{\left(-\frac{3913}{625}\right)^{2}-4\times \frac{100601}{10000}\left(-\frac{667449}{625}\right)}}{2\times \frac{100601}{10000}}$$

Square 1\times \frac{26}{25}\left(-\frac{301}{100}\right)\times 2.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\sqrt{\frac{15311569}{390625}-4\times \frac{100601}{10000}\left(-\frac{667449}{625}\right)}}{2\times \frac{100601}{10000}}$$

Multiply -4 times 1+1\left(-\frac{301}{100}\right)^{2}.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\sqrt{\frac{15311569}{390625}-\frac{100601}{2500}\left(-\frac{667449}{625}\right)}}{2\times \frac{100601}{10000}}$$

Multiply -\frac{100601}{2500} times -\frac{667449}{625} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\sqrt{\frac{15311569}{390625}+\frac{67146036849}{1562500}}}{2\times \frac{100601}{10000}}$$

Add \frac{15311569}{390625} to \frac{67146036849}{1562500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\sqrt{\frac{107531653}{2500}}}{2\times \frac{100601}{10000}}$$

Take the square root of \frac{107531653}{2500}.

$$x=\frac{-\left(-\frac{3913}{625}\right)±\frac{\sqrt{107531653}}{50}}{2\times \frac{100601}{10000}}$$

The opposite of 1\times \frac{26}{25}\left(-\frac{301}{100}\right)\times 2 is \frac{3913}{625}.

$$x=\frac{\frac{3913}{625}±\frac{\sqrt{107531653}}{50}}{2\times \frac{100601}{10000}}$$

Multiply 2 times 1+1\left(-\frac{301}{100}\right)^{2}.

$$x=\frac{\frac{3913}{625}±\frac{\sqrt{107531653}}{50}}{\frac{100601}{5000}}$$

Now solve the equation x=\frac{\frac{3913}{625}±\frac{\sqrt{107531653}}{50}}{\frac{100601}{5000}} when ± is plus. Add \frac{3913}{625} to \frac{\sqrt{107531653}}{50}.

$$x=\frac{\frac{\sqrt{107531653}}{50}+\frac{3913}{625}}{\frac{100601}{5000}}$$

Divide \frac{3913}{625}+\frac{\sqrt{107531653}}{50} by \frac{100601}{5000} by multiplying \frac{3913}{625}+\frac{\sqrt{107531653}}{50} by the reciprocal of \frac{100601}{5000}.

$$x=\frac{100\sqrt{107531653}+31304}{100601}$$

Now solve the equation x=\frac{\frac{3913}{625}±\frac{\sqrt{107531653}}{50}}{\frac{100601}{5000}} when ± is minus. Subtract \frac{\sqrt{107531653}}{50} from \frac{3913}{625}.

$$x=\frac{-\frac{\sqrt{107531653}}{50}+\frac{3913}{625}}{\frac{100601}{5000}}$$

Divide \frac{3913}{625}-\frac{\sqrt{107531653}}{50} by \frac{100601}{5000} by multiplying \frac{3913}{625}-\frac{\sqrt{107531653}}{50} by the reciprocal of \frac{100601}{5000}.

$$x=\frac{31304-100\sqrt{107531653}}{100601}$$

There are two solutions for x: \frac{31304+100\sqrt{107531653}}{100601} and \frac{31304-100\sqrt{107531653}}{100601}. Substitute \frac{31304+100\sqrt{107531653}}{100601} for x in the equation y=-\frac{301}{100}x+\frac{26}{25} to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{301}{100}\times \frac{100\sqrt{107531653}+31304}{100601}+\frac{26}{25}$$

Multiply -\frac{301}{100} times \frac{31304+100\sqrt{107531653}}{100601}.

$$y=-\frac{301\times \frac{100\sqrt{107531653}+31304}{100601}}{100}+\frac{26}{25}$$

Now substitute \frac{31304-100\sqrt{107531653}}{100601} for x in the equation y=-\frac{301}{100}x+\frac{26}{25} and solve to find the corresponding solution for y that satisfies both equations.

$$y=-\frac{301}{100}\times \frac{31304-100\sqrt{107531653}}{100601}+\frac{26}{25}$$

Multiply -\frac{301}{100} times \frac{31304-100\sqrt{107531653}}{100601}.

$$y=-\frac{301\times \frac{31304-100\sqrt{107531653}}{100601}}{100}+\frac{26}{25}$$

The system is now solved.

$$y=-\frac{301\times \frac{100\sqrt{107531653}+31304}{100601}}{100}+\frac{26}{25},x=\frac{100\sqrt{107531653}+31304}{100601}\text{ or }y=-\frac{301\times \frac{31304-100\sqrt{107531653}}{100601}}{100}+\frac{26}{25},x=\frac{31304-100\sqrt{107531653}}{100601}$$