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x+y=1,y^{2}+x^{2}=4
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=1
Solve x+y=1 for x by isolating x on the left hand side of the equal sign.
x=-y+1
Subtract y from both sides of the equation.
y^{2}+\left(-y+1\right)^{2}=4
Substitute -y+1 for x in the other equation, y^{2}+x^{2}=4.
y^{2}+y^{2}-2y+1=4
Square -y+1.
2y^{2}-2y+1=4
Add y^{2} to y^{2}.
2y^{2}-2y-3=0
Subtract 4 from both sides of the equation.
y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-3\right)}}{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 1\left(-1\right)\times 2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-3\right)}}{2\times 2}
Square 1\times 1\left(-1\right)\times 2.
y=\frac{-\left(-2\right)±\sqrt{4-8\left(-3\right)}}{2\times 2}
Multiply -4 times 1+1\left(-1\right)^{2}.
y=\frac{-\left(-2\right)±\sqrt{4+24}}{2\times 2}
Multiply -8 times -3.
y=\frac{-\left(-2\right)±\sqrt{28}}{2\times 2}
Add 4 to 24.
y=\frac{-\left(-2\right)±2\sqrt{7}}{2\times 2}
Take the square root of 28.
y=\frac{2±2\sqrt{7}}{2\times 2}
The opposite of 1\times 1\left(-1\right)\times 2 is 2.
y=\frac{2±2\sqrt{7}}{4}
Multiply 2 times 1+1\left(-1\right)^{2}.
y=\frac{2\sqrt{7}+2}{4}
Now solve the equation y=\frac{2±2\sqrt{7}}{4} when ± is plus. Add 2 to 2\sqrt{7}.
y=\frac{\sqrt{7}+1}{2}
Divide 2+2\sqrt{7} by 4.
y=\frac{2-2\sqrt{7}}{4}
Now solve the equation y=\frac{2±2\sqrt{7}}{4} when ± is minus. Subtract 2\sqrt{7} from 2.
y=\frac{1-\sqrt{7}}{2}
Divide 2-2\sqrt{7} by 4.
x=-\frac{\sqrt{7}+1}{2}+1
There are two solutions for y: \frac{1+\sqrt{7}}{2} and \frac{1-\sqrt{7}}{2}. Substitute \frac{1+\sqrt{7}}{2} for y in the equation x=-y+1 to find the corresponding solution for x that satisfies both equations.
x=-\frac{1-\sqrt{7}}{2}+1
Now substitute \frac{1-\sqrt{7}}{2} for y in the equation x=-y+1 and solve to find the corresponding solution for x that satisfies both equations.
x=-\frac{\sqrt{7}+1}{2}+1,y=\frac{\sqrt{7}+1}{2}\text{ or }x=-\frac{1-\sqrt{7}}{2}+1,y=\frac{1-\sqrt{7}}{2}
The system is now solved.