Solve {l}{2x-5y=7}{12x^2-5y^2=7}

$$\left\{ \begin{array} { l } { 2 x - 5 y = 7 } \\ { 12 x ^ { 2 } - 5 y ^ { 2 } = 7 } \end{array} \right.$$

Solve for x, y

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

$$x=-\frac{3}{2}=-1.5\text{, }y=-2$$

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.

$$2x-5y=7,-5y^{2}+12x^{2}=7$$

Solve 2x-5y=7 for x by isolating x on the left hand side of the equal sign.

$$2x-5y=7$$

Subtract -5y from both sides of the equation.

$$2x=5y+7$$

Divide both sides by 2.

$$x=\frac{5}{2}y+\frac{7}{2}$$

Substitute \frac{5}{2}y+\frac{7}{2} for x in the other equation, -5y^{2}+12x^{2}=7.

$$-5y^{2}+12\left(\frac{5}{2}y+\frac{7}{2}\right)^{2}=7$$

Square \frac{5}{2}y+\frac{7}{2}.

$$-5y^{2}+12\left(\frac{25}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}\right)=7$$

Multiply 12 times \frac{25}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}.

$$-5y^{2}+75y^{2}+210y+147=7$$

Add -5y^{2} to 75y^{2}.

$$70y^{2}+210y+147=7$$

Subtract 7 from both sides of the equation.

$$70y^{2}+210y+140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5+12\times \left(\frac{5}{2}\right)^{2} for a, 12\times \frac{7}{2}\times 2\times \frac{5}{2} for b, and 140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-210±\sqrt{210^{2}-4\times 70\times 140}}{2\times 70}$$

Square 12\times \frac{7}{2}\times 2\times \frac{5}{2}.

$$y=\frac{-210±\sqrt{44100-4\times 70\times 140}}{2\times 70}$$

Multiply -4 times -5+12\times \left(\frac{5}{2}\right)^{2}.

$$y=\frac{-210±\sqrt{44100-280\times 140}}{2\times 70}$$

Multiply -280 times 140.

$$y=\frac{-210±\sqrt{44100-39200}}{2\times 70}$$

Add 44100 to -39200.

$$y=\frac{-210±\sqrt{4900}}{2\times 70}$$

Take the square root of 4900.

$$y=\frac{-210±70}{2\times 70}$$

Multiply 2 times -5+12\times \left(\frac{5}{2}\right)^{2}.

$$y=\frac{-210±70}{140}$$

Now solve the equation y=\frac{-210±70}{140} when ± is plus. Add -210 to 70.

$$y=-\frac{140}{140}$$

Divide -140 by 140.

$$y=-1$$

Now solve the equation y=\frac{-210±70}{140} when ± is minus. Subtract 70 from -210.

$$y=-\frac{280}{140}$$

Divide -280 by 140.

$$y=-2$$

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

$$x=\frac{5}{2}\left(-1\right)+\frac{7}{2}$$

Multiply \frac{5}{2} times -1.

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

Add -\frac{5}{2} to \frac{7}{2}.

$$x=1$$

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

$$x=\frac{5}{2}\left(-2\right)+\frac{7}{2}$$

Multiply \frac{5}{2} times -2.

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

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

$$x=-\frac{3}{2}$$

The system is now solved.

$$x=1,y=-1\text{ or }x=-\frac{3}{2},y=-2$$