Solve {l}{2x^2-y^2=17}{x+2y=1}

$$\left. \begin{array}{l}{ 2 x ^ { 2 } - y ^ { 2 } = 17 }\\{ x + 2 y = 1 }\end{array} \right.$$

Solve for x, y

$$x=3\text{, }y=-1$$

$$x=-\frac{23}{7}\approx -3.285714286\text{, }y=\frac{15}{7}\approx 2.142857143$$

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.

$$x+2y=1,-y^{2}+2x^{2}=17$$

Solve x+2y=1 for x by isolating x on the left hand side of the equal sign.

$$x+2y=1$$

Subtract 2y from both sides of the equation.

$$x=-2y+1$$

Substitute -2y+1 for x in the other equation, -y^{2}+2x^{2}=17.

$$-y^{2}+2\left(-2y+1\right)^{2}=17$$

Square -2y+1.

$$-y^{2}+2\left(4y^{2}-4y+1\right)=17$$

Multiply 2 times 4y^{2}-4y+1.

$$-y^{2}+8y^{2}-8y+2=17$$

Add -y^{2} to 8y^{2}.

$$7y^{2}-8y+2=17$$

Subtract 17 from both sides of the equation.

$$7y^{2}-8y-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1+2\left(-2\right)^{2} for a, 2\times 1\left(-2\right)\times 2 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 7\left(-15\right)}}{2\times 7}$$

Square 2\times 1\left(-2\right)\times 2.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 7\left(-15\right)}}{2\times 7}$$

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

$$y=\frac{-\left(-8\right)±\sqrt{64-28\left(-15\right)}}{2\times 7}$$

Multiply -28 times -15.

$$y=\frac{-\left(-8\right)±\sqrt{64+420}}{2\times 7}$$

Add 64 to 420.

$$y=\frac{-\left(-8\right)±\sqrt{484}}{2\times 7}$$

Take the square root of 484.

$$y=\frac{-\left(-8\right)±22}{2\times 7}$$

The opposite of 2\times 1\left(-2\right)\times 2 is 8.

$$y=\frac{8±22}{2\times 7}$$

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

$$y=\frac{8±22}{14}$$

Now solve the equation y=\frac{8±22}{14} when ± is plus. Add 8 to 22.

$$y=\frac{30}{14}$$

Reduce the fraction \frac{30}{14} to lowest terms by extracting and canceling out 2.

$$y=\frac{15}{7}$$

Now solve the equation y=\frac{8±22}{14} when ± is minus. Subtract 22 from 8.

$$y=-\frac{14}{14}$$

Divide -14 by 14.

$$y=-1$$

There are two solutions for y: \frac{15}{7} and -1. Substitute \frac{15}{7} for y in the equation x=-2y+1 to find the corresponding solution for x that satisfies both equations.

$$x=-2\times \frac{15}{7}+1$$

Multiply -2 times \frac{15}{7}.

$$x=-\frac{30}{7}+1$$

Add -2\times \frac{15}{7} to 1.

$$x=-\frac{23}{7}$$

Now substitute -1 for y in the equation x=-2y+1 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-2\left(-1\right)+1$$

Multiply -2 times -1.

$$x=2+1$$

Add -2\left(-1\right) to 1.

$$x=3$$

The system is now solved.

$$x=-\frac{23}{7},y=\frac{15}{7}\text{ or }x=3,y=-1$$