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x+y=5,y^{2}+x^{2}=16
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=5
Solve x+y=5 for x by isolating x on the left hand side of the equal sign.
x=-y+5
Subtract y from both sides of the equation.
y^{2}+\left(-y+5\right)^{2}=16
Substitute -y+5 for x in the other equation, y^{2}+x^{2}=16.
y^{2}+y^{2}-10y+25=16
Square -y+5.
2y^{2}-10y+25=16
Add y^{2} to y^{2}.
2y^{2}-10y+9=0
Subtract 16 from both sides of the equation.
y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 2\times 9}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\times 5\left(-1\right)\times 2 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-10\right)±\sqrt{100-4\times 2\times 9}}{2\times 2}
Square 1\times 5\left(-1\right)\times 2.
y=\frac{-\left(-10\right)±\sqrt{100-8\times 9}}{2\times 2}
Multiply -4 times 1+1\left(-1\right)^{2}.
y=\frac{-\left(-10\right)±\sqrt{100-72}}{2\times 2}
Multiply -8 times 9.
y=\frac{-\left(-10\right)±\sqrt{28}}{2\times 2}
Add 100 to -72.
y=\frac{-\left(-10\right)±2\sqrt{7}}{2\times 2}
Take the square root of 28.
y=\frac{10±2\sqrt{7}}{2\times 2}
The opposite of 1\times 5\left(-1\right)\times 2 is 10.
y=\frac{10±2\sqrt{7}}{4}
Multiply 2 times 1+1\left(-1\right)^{2}.
y=\frac{2\sqrt{7}+10}{4}
Now solve the equation y=\frac{10±2\sqrt{7}}{4} when ± is plus. Add 10 to 2\sqrt{7}.
y=\frac{\sqrt{7}+5}{2}
Divide 10+2\sqrt{7} by 4.
y=\frac{10-2\sqrt{7}}{4}
Now solve the equation y=\frac{10±2\sqrt{7}}{4} when ± is minus. Subtract 2\sqrt{7} from 10.
y=\frac{5-\sqrt{7}}{2}
Divide 10-2\sqrt{7} by 4.
x=-\frac{\sqrt{7}+5}{2}+5
There are two solutions for y: \frac{5+\sqrt{7}}{2} and \frac{5-\sqrt{7}}{2}. Substitute \frac{5+\sqrt{7}}{2} for y in the equation x=-y+5 to find the corresponding solution for x that satisfies both equations.
x=-\frac{5-\sqrt{7}}{2}+5
Now substitute \frac{5-\sqrt{7}}{2} for y in the equation x=-y+5 and solve to find the corresponding solution for x that satisfies both equations.
x=-\frac{\sqrt{7}+5}{2}+5,y=\frac{\sqrt{7}+5}{2}\text{ or }x=-\frac{5-\sqrt{7}}{2}+5,y=\frac{5-\sqrt{7}}{2}
The system is now solved.