Solve {l}{sqrt{5}x+5y=-25}{sqrt{20}x-3y=15}

$$\left. \begin{array} { l } { \sqrt { 5 } x + 5 y = - 25 } \\ { \sqrt { 20 } x - 3 y = 15 } \end{array} \right.$$

Solve for x, y

$$x=0$$

$$y=-5$$

Steps Using Substitution

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.

$$\sqrt{5}x+5y=-25,\sqrt{20}x-3y=15$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$\sqrt{5}x+5y=-25$$

Subtract 5y from both sides of the equation.

$$\sqrt{5}x=-5y-25$$

Divide both sides by \sqrt{5}.

$$x=\frac{\sqrt{5}}{5}\left(-5y-25\right)$$

Multiply \frac{\sqrt{5}}{5} times -5y-25.

$$x=\left(-\sqrt{5}\right)y-5\sqrt{5}$$

Substitute -\left(5+y\right)\sqrt{5} for x in the other equation, \sqrt{20}x-3y=15.

$$\sqrt{20}\left(\left(-\sqrt{5}\right)y-5\sqrt{5}\right)-3y=15$$

Multiply \sqrt{20} times -\left(5+y\right)\sqrt{5}.

$$-10y-50-3y=15$$

Add -10y to -3y.

$$-13y-50=15$$

Add 50 to both sides of the equation.

$$-13y=65$$

Divide both sides by -13.

$$y=-5$$

Substitute -5 for y in x=\left(-\sqrt{5}\right)y-5\sqrt{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-\sqrt{5}\right)\left(-5\right)-5\sqrt{5}$$

Multiply -\sqrt{5} times -5.

$$x=5\sqrt{5}-5\sqrt{5}$$

Add -5\sqrt{5} to 5\sqrt{5}.

$$x=0$$

The system is now solved.

$$x=0,y=-5$$

Steps Using Elimination

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$\sqrt{5}x+5y=-25,\sqrt{20}x-3y=15$$

To make \sqrt{5}x and 2\sqrt{5}x equal, multiply all terms on each side of the first equation by \sqrt{20} and all terms on each side of the second by \sqrt{5}.

$$\sqrt{20}\sqrt{5}x+\sqrt{20}\times 5y=\sqrt{20}\left(-25\right),\sqrt{5}\sqrt{20}x+\sqrt{5}\left(-3\right)y=\sqrt{5}\times 15$$

Simplify.

$$10x+10\sqrt{5}y=-50\sqrt{5},10x+\left(-3\sqrt{5}\right)y=15\sqrt{5}$$

Subtract 10x+\left(-3\sqrt{5}\right)y=15\sqrt{5} from 10x+10\sqrt{5}y=-50\sqrt{5} by subtracting like terms on each side of the equal sign.

$$10x-10x+10\sqrt{5}y+3\sqrt{5}y=-50\sqrt{5}-15\sqrt{5}$$

Add 10x to -10x. Terms 10x and -10x cancel out, leaving an equation with only one variable that can be solved.

$$10\sqrt{5}y+3\sqrt{5}y=-50\sqrt{5}-15\sqrt{5}$$

Add 10\sqrt{5}y to 3\sqrt{5}y.

$$13\sqrt{5}y=-50\sqrt{5}-15\sqrt{5}$$

Add -50\sqrt{5} to -15\sqrt{5}.

$$13\sqrt{5}y=-65\sqrt{5}$$

Divide both sides by 13\sqrt{5}.

$$y=-5$$

Substitute -5 for y in \sqrt{20}x-3y=15. Because the resulting equation contains only one variable, you can solve for x directly.

$$\sqrt{20}x-3\left(-5\right)=15$$

Multiply -3 times -5.

$$\sqrt{20}x+15=15$$

Subtract 15 from both sides of the equation.

$$\sqrt{20}x=0$$

Divide both sides by \sqrt{20}.

$$x=0$$

The system is now solved.

$$x=0,y=-5$$