Solve {l}{-(x+y)=7}{x^2+y^2=25}

$$\left. \begin{array} { l } { - ( x + y ) = 7 } \\ { x ^ { 2 } + y ^ { 2 } = 25 } \end{array} \right.$$

Solve for x, y

$$x=-3\text{, }y=-4$$

$$x=-4\text{, }y=-3$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. To find the opposite of x+y, find the opposite of each term.

$$-x-y=7$$

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.

$$-x-y=7,y^{2}+x^{2}=25$$

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

$$-x-y=7$$

Subtract -y from both sides of the equation.

$$-x=y+7$$

Divide both sides by -1.

$$x=-y-7$$

Substitute -y-7 for x in the other equation, y^{2}+x^{2}=25.

$$y^{2}+\left(-y-7\right)^{2}=25$$

Square -y-7.

$$y^{2}+y^{2}+14y+49=25$$

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

$$2y^{2}+14y+49=25$$

Subtract 25 from both sides of the equation.

$$2y^{2}+14y+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\left(-7\right)\left(-1\right)\times 2 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-14±\sqrt{14^{2}-4\times 2\times 24}}{2\times 2}$$

Square 1\left(-7\right)\left(-1\right)\times 2.

$$y=\frac{-14±\sqrt{196-4\times 2\times 24}}{2\times 2}$$

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

$$y=\frac{-14±\sqrt{196-8\times 24}}{2\times 2}$$

Multiply -8 times 24.

$$y=\frac{-14±\sqrt{196-192}}{2\times 2}$$

Add 196 to -192.

$$y=\frac{-14±\sqrt{4}}{2\times 2}$$

Take the square root of 4.

$$y=\frac{-14±2}{2\times 2}$$

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

$$y=\frac{-14±2}{4}$$

Now solve the equation y=\frac{-14±2}{4} when ± is plus. Add -14 to 2.

$$y=-\frac{12}{4}$$

Divide -12 by 4.

$$y=-3$$

Now solve the equation y=\frac{-14±2}{4} when ± is minus. Subtract 2 from -14.

$$y=-\frac{16}{4}$$

Divide -16 by 4.

$$y=-4$$

There are two solutions for y: -3 and -4. Substitute -3 for y in the equation x=-y-7 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-3\right)-7$$

Multiply -1 times -3.

$$x=3-7$$

Add -3\left(-1\right) to -7.

$$x=-4$$

Now substitute -4 for y in the equation x=-y-7 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-4\right)-7$$

Multiply -1 times -4.

$$x=4-7$$

Add -4\left(-1\right) to -7.

$$x=-3$$

The system is now solved.

$$x=-4,y=-3\text{ or }x=-3,y=-4$$