Solve {l}{x^2+y^2=169}{x+y=17}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 169 } \\ { x + y = 17 } \end{array} \right.$$

Solve for x, y

$$x=12\text{, }y=5$$

$$x=5\text{, }y=12$$

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.

$$x+y=17,y^{2}+x^{2}=169$$

Solve x+y=17 for x by isolating x on the left hand side of the equal sign.

$$x+y=17$$

Subtract y from both sides of the equation.

$$x=-y+17$$

Substitute -y+17 for x in the other equation, y^{2}+x^{2}=169.

$$y^{2}+\left(-y+17\right)^{2}=169$$

Square -y+17.

$$y^{2}+y^{2}-34y+289=169$$

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

$$2y^{2}-34y+289=169$$

Subtract 169 from both sides of the equation.

$$2y^{2}-34y+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 17\left(-1\right)\times 2 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 120}}{2\times 2}$$

Square 1\times 17\left(-1\right)\times 2.

$$y=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 120}}{2\times 2}$$

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

$$y=\frac{-\left(-34\right)±\sqrt{1156-8\times 120}}{2\times 2}$$

Multiply -8 times 120.

$$y=\frac{-\left(-34\right)±\sqrt{1156-960}}{2\times 2}$$

Add 1156 to -960.

$$y=\frac{-\left(-34\right)±\sqrt{196}}{2\times 2}$$

Take the square root of 196.

$$y=\frac{-\left(-34\right)±14}{2\times 2}$$

The opposite of 1\times 17\left(-1\right)\times 2 is 34.

$$y=\frac{34±14}{2\times 2}$$

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

$$y=\frac{34±14}{4}$$

Now solve the equation y=\frac{34±14}{4} when ± is plus. Add 34 to 14.

$$y=\frac{48}{4}$$

Divide 48 by 4.

$$y=12$$

Now solve the equation y=\frac{34±14}{4} when ± is minus. Subtract 14 from 34.

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

Divide 20 by 4.

$$y=5$$

There are two solutions for y: 12 and 5. Substitute 12 for y in the equation x=-y+17 to find the corresponding solution for x that satisfies both equations.

$$x=-12+17$$

Add -12 to 17.

$$x=5$$

Now substitute 5 for y in the equation x=-y+17 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-5+17$$

Add -5 to 17.

$$x=12$$

The system is now solved.

$$x=5,y=12\text{ or }x=12,y=5$$