Solve {l}{3x^2+2y^2=2}{2x+7y=3}

$$\left\{ \begin{array} { l } { 3 x ^ { 2 } + 2 y ^ { 2 } = 2 } \\ { 2 x + 7 y = 3 } \end{array} \right.$$

Solve for x, y

$$x=\frac{4}{5}=0.8\text{, }y=\frac{1}{5}=0.2$$

$$x=-\frac{20}{31}\approx -0.64516129\text{, }y=\frac{19}{31}\approx 0.612903226$$

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.

$$2x+7y=3,2y^{2}+3x^{2}=2$$

Solve 2x+7y=3 for x by isolating x on the left hand side of the equal sign.

$$2x+7y=3$$

Subtract 7y from both sides of the equation.

$$2x=-7y+3$$

Divide both sides by 2.

$$x=-\frac{7}{2}y+\frac{3}{2}$$

Substitute -\frac{7}{2}y+\frac{3}{2} for x in the other equation, 2y^{2}+3x^{2}=2.

$$2y^{2}+3\left(-\frac{7}{2}y+\frac{3}{2}\right)^{2}=2$$

Square -\frac{7}{2}y+\frac{3}{2}.

$$2y^{2}+3\left(\frac{49}{4}y^{2}-\frac{21}{2}y+\frac{9}{4}\right)=2$$

Multiply 3 times \frac{49}{4}y^{2}-\frac{21}{2}y+\frac{9}{4}.

$$2y^{2}+\frac{147}{4}y^{2}-\frac{63}{2}y+\frac{27}{4}=2$$

Add 2y^{2} to \frac{147}{4}y^{2}.

$$\frac{155}{4}y^{2}-\frac{63}{2}y+\frac{27}{4}=2$$

Subtract 2 from both sides of the equation.

$$\frac{155}{4}y^{2}-\frac{63}{2}y+\frac{19}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2+3\left(-\frac{7}{2}\right)^{2} for a, 3\times \frac{3}{2}\left(-\frac{7}{2}\right)\times 2 for b, and \frac{19}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{63}{2}\right)±\sqrt{\left(-\frac{63}{2}\right)^{2}-4\times \frac{155}{4}\times \frac{19}{4}}}{2\times \frac{155}{4}}$$

Square 3\times \frac{3}{2}\left(-\frac{7}{2}\right)\times 2.

$$y=\frac{-\left(-\frac{63}{2}\right)±\sqrt{\frac{3969}{4}-4\times \frac{155}{4}\times \frac{19}{4}}}{2\times \frac{155}{4}}$$

Multiply -4 times 2+3\left(-\frac{7}{2}\right)^{2}.

$$y=\frac{-\left(-\frac{63}{2}\right)±\sqrt{\frac{3969}{4}-155\times \frac{19}{4}}}{2\times \frac{155}{4}}$$

Multiply -155 times \frac{19}{4}.

$$y=\frac{-\left(-\frac{63}{2}\right)±\sqrt{\frac{3969-2945}{4}}}{2\times \frac{155}{4}}$$

Add \frac{3969}{4} to -\frac{2945}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{63}{2}\right)±\sqrt{256}}{2\times \frac{155}{4}}$$

Take the square root of 256.

$$y=\frac{-\left(-\frac{63}{2}\right)±16}{2\times \frac{155}{4}}$$

The opposite of 3\times \frac{3}{2}\left(-\frac{7}{2}\right)\times 2 is \frac{63}{2}.

$$y=\frac{\frac{63}{2}±16}{2\times \frac{155}{4}}$$

Multiply 2 times 2+3\left(-\frac{7}{2}\right)^{2}.

$$y=\frac{\frac{63}{2}±16}{\frac{155}{2}}$$

Now solve the equation y=\frac{\frac{63}{2}±16}{\frac{155}{2}} when ± is plus. Add \frac{63}{2} to 16.

$$y=\frac{\frac{95}{2}}{\frac{155}{2}}$$

Divide \frac{95}{2} by \frac{155}{2} by multiplying \frac{95}{2} by the reciprocal of \frac{155}{2}.

$$y=\frac{19}{31}$$

Now solve the equation y=\frac{\frac{63}{2}±16}{\frac{155}{2}} when ± is minus. Subtract 16 from \frac{63}{2}.

$$y=\frac{\frac{31}{2}}{\frac{155}{2}}$$

Divide \frac{31}{2} by \frac{155}{2} by multiplying \frac{31}{2} by the reciprocal of \frac{155}{2}.

$$y=\frac{1}{5}$$

There are two solutions for y: \frac{19}{31} and \frac{1}{5}. Substitute \frac{19}{31} for y in the equation x=-\frac{7}{2}y+\frac{3}{2} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{7}{2}\times \frac{19}{31}+\frac{3}{2}$$

Multiply -\frac{7}{2} times \frac{19}{31} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{133}{62}+\frac{3}{2}$$

Add -\frac{7}{2}\times \frac{19}{31} to \frac{3}{2}.

$$x=-\frac{20}{31}$$

Now substitute \frac{1}{5} for y in the equation x=-\frac{7}{2}y+\frac{3}{2} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{7}{2}\times \frac{1}{5}+\frac{3}{2}$$

Multiply -\frac{7}{2} times \frac{1}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{7}{10}+\frac{3}{2}$$

Add -\frac{7}{2}\times \frac{1}{5} to \frac{3}{2}.

$$x=\frac{4}{5}$$

The system is now solved.

$$x=-\frac{20}{31},y=\frac{19}{31}\text{ or }x=\frac{4}{5},y=\frac{1}{5}$$