Solve {l}{x^2+y^2=74}{x+y=12}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 74 } \\ { x + y = 12 } \end{array} \right.$$

Solve for x, y

$$x=7\text{, }y=5$$

$$x=5\text{, }y=7$$

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.

$$x+y=12,y^{2}+x^{2}=74$$

Solve x+y=12 for x by isolating x on the left hand side of the equal sign.

$$x+y=12$$

Subtract y from both sides of the equation.

$$x=-y+12$$

Substitute -y+12 for x in the other equation, y^{2}+x^{2}=74.

$$y^{2}+\left(-y+12\right)^{2}=74$$

Square -y+12.

$$y^{2}+y^{2}-24y+144=74$$

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

$$2y^{2}-24y+144=74$$

Subtract 74 from both sides of the equation.

$$2y^{2}-24y+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 12\left(-1\right)\times 2 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\times 70}}{2\times 2}$$

Square 1\times 12\left(-1\right)\times 2.

$$y=\frac{-\left(-24\right)±\sqrt{576-4\times 2\times 70}}{2\times 2}$$

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

$$y=\frac{-\left(-24\right)±\sqrt{576-8\times 70}}{2\times 2}$$

Multiply -8 times 70.

$$y=\frac{-\left(-24\right)±\sqrt{576-560}}{2\times 2}$$

Add 576 to -560.

$$y=\frac{-\left(-24\right)±\sqrt{16}}{2\times 2}$$

Take the square root of 16.

$$y=\frac{-\left(-24\right)±4}{2\times 2}$$

The opposite of 1\times 12\left(-1\right)\times 2 is 24.

$$y=\frac{24±4}{2\times 2}$$

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

$$y=\frac{24±4}{4}$$

Now solve the equation y=\frac{24±4}{4} when ± is plus. Add 24 to 4.

$$y=\frac{28}{4}$$

Divide 28 by 4.

$$y=7$$

Now solve the equation y=\frac{24±4}{4} when ± is minus. Subtract 4 from 24.

$$y=\frac{20}{4}$$

Divide 20 by 4.

$$y=5$$

There are two solutions for y: 7 and 5. Substitute 7 for y in the equation x=-y+12 to find the corresponding solution for x that satisfies both equations.

$$x=-7+12$$

Add -7 to 12.

$$x=5$$

Now substitute 5 for y in the equation x=-y+12 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-5+12$$

Add -5 to 12.

$$x=7$$

The system is now solved.

$$x=5,y=7\text{ or }x=7,y=5$$