Solve {l}{2x+4y=11}{17x^2+3y^2=15}

$$\left. \begin{array} { l } { 2 x + 4 y = 11 } \\ { 17 x ^ { 2 } + 3 y ^ { 2 } = 15 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{7\sqrt{39}i}{71}+\frac{33}{142}\approx 0.232394366+0.615704028i\text{, }y=-\frac{7\sqrt{39}i}{142}+\frac{187}{71}\approx 2.633802817-0.307852014i$$

$$x=-\frac{7\sqrt{39}i}{71}+\frac{33}{142}\approx 0.232394366-0.615704028i\text{, }y=\frac{7\sqrt{39}i}{142}+\frac{187}{71}\approx 2.633802817+0.307852014i$$

Steps Using Substitution and Quadratic Formula

Solve 2x+4y=11 for x by isolating x on the left hand side of the equal sign.

$$2x+4y=11$$

Subtract 4y from both sides of the equation.

$$2x=-4y+11$$

Divide both sides by 2.

$$x=-2y+\frac{11}{2}$$

Substitute -2y+\frac{11}{2} for x in the other equation, 3y^{2}+17x^{2}=15.

$$3y^{2}+17\left(-2y+\frac{11}{2}\right)^{2}=15$$

Square -2y+\frac{11}{2}.

$$3y^{2}+17\left(4y^{2}-22y+\frac{121}{4}\right)=15$$

Multiply 17 times 4y^{2}-22y+\frac{121}{4}.

$$3y^{2}+68y^{2}-374y+\frac{2057}{4}=15$$

Add 3y^{2} to 68y^{2}.

$$71y^{2}-374y+\frac{2057}{4}=15$$

Subtract 15 from both sides of the equation.

$$71y^{2}-374y+\frac{1997}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+17\left(-2\right)^{2} for a, 17\times \frac{11}{2}\left(-2\right)\times 2 for b, and \frac{1997}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-374\right)±\sqrt{\left(-374\right)^{2}-4\times 71\times \frac{1997}{4}}}{2\times 71}$$

Square 17\times \frac{11}{2}\left(-2\right)\times 2.

$$y=\frac{-\left(-374\right)±\sqrt{139876-4\times 71\times \frac{1997}{4}}}{2\times 71}$$

Multiply -4 times 3+17\left(-2\right)^{2}.

$$y=\frac{-\left(-374\right)±\sqrt{139876-284\times \frac{1997}{4}}}{2\times 71}$$

Multiply -284 times \frac{1997}{4}.

$$y=\frac{-\left(-374\right)±\sqrt{139876-141787}}{2\times 71}$$

Add 139876 to -141787.

$$y=\frac{-\left(-374\right)±\sqrt{-1911}}{2\times 71}$$

Take the square root of -1911.

$$y=\frac{-\left(-374\right)±7\sqrt{39}i}{2\times 71}$$

The opposite of 17\times \frac{11}{2}\left(-2\right)\times 2 is 374.

$$y=\frac{374±7\sqrt{39}i}{2\times 71}$$

Multiply 2 times 3+17\left(-2\right)^{2}.

$$y=\frac{374±7\sqrt{39}i}{142}$$

Now solve the equation y=\frac{374±7\sqrt{39}i}{142} when ± is plus. Add 374 to 7i\sqrt{39}.

$$y=\frac{374+7\sqrt{39}i}{142}$$

Divide 374+7i\sqrt{39} by 142.

$$y=\frac{7\sqrt{39}i}{142}+\frac{187}{71}$$

Now solve the equation y=\frac{374±7\sqrt{39}i}{142} when ± is minus. Subtract 7i\sqrt{39} from 374.

$$y=\frac{-7\sqrt{39}i+374}{142}$$

Divide 374-7i\sqrt{39} by 142.

$$y=-\frac{7\sqrt{39}i}{142}+\frac{187}{71}$$

There are two solutions for y: \frac{187}{71}+\frac{7i\sqrt{39}}{142} and \frac{187}{71}-\frac{7i\sqrt{39}}{142}. Substitute \frac{187}{71}+\frac{7i\sqrt{39}}{142} for y in the equation x=-2y+\frac{11}{2} to find the corresponding solution for x that satisfies both equations.

$$x=-2\left(\frac{7\sqrt{39}i}{142}+\frac{187}{71}\right)+\frac{11}{2}$$

Now substitute \frac{187}{71}-\frac{7i\sqrt{39}}{142} for y in the equation x=-2y+\frac{11}{2} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-2\left(-\frac{7\sqrt{39}i}{142}+\frac{187}{71}\right)+\frac{11}{2}$$

The system is now solved.

$$x=-2\left(\frac{7\sqrt{39}i}{142}+\frac{187}{71}\right)+\frac{11}{2},y=\frac{7\sqrt{39}i}{142}+\frac{187}{71}\text{ or }x=-2\left(-\frac{7\sqrt{39}i}{142}+\frac{187}{71}\right)+\frac{11}{2},y=-\frac{7\sqrt{39}i}{142}+\frac{187}{71}$$