Solve for x, y
x=-\frac{\sqrt{2}}{2}\approx -0.707106781\text{, }y=-\frac{\sqrt{2}}{2}\approx -0.707106781
x=\frac{\sqrt{2}}{2}\approx 0.707106781\text{, }y=\frac{\sqrt{2}}{2}\approx 0.707106781
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x-y=0
Consider the second equation. Subtract y from both sides.
x-y=0,y^{2}+x^{2}=1
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=0
Solve x-y=0 for x by isolating x on the left hand side of the equal sign.
x=y
Subtract -y from both sides of the equation.
y^{2}+y^{2}=1
Substitute y for x in the other equation, y^{2}+x^{2}=1.
2y^{2}=1
Add y^{2} to y^{2}.
2y^{2}-1=0
Subtract 1 from both sides of the equation.
y=\frac{0±\sqrt{0^{2}-4\times 2\left(-1\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times 1^{2} for a, 1\times 0\times 1\times 2 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{0±\sqrt{-4\times 2\left(-1\right)}}{2\times 2}
Square 1\times 0\times 1\times 2.
y=\frac{0±\sqrt{-8\left(-1\right)}}{2\times 2}
Multiply -4 times 1+1\times 1^{2}.
y=\frac{0±\sqrt{8}}{2\times 2}
Multiply -8 times -1.
y=\frac{0±2\sqrt{2}}{2\times 2}
Take the square root of 8.
y=\frac{0±2\sqrt{2}}{4}
Multiply 2 times 1+1\times 1^{2}.
y=\frac{\sqrt{2}}{2}
Now solve the equation y=\frac{0±2\sqrt{2}}{4} when ± is plus.
y=-\frac{\sqrt{2}}{2}
Now solve the equation y=\frac{0±2\sqrt{2}}{4} when ± is minus.
x=\frac{\sqrt{2}}{2}
There are two solutions for y: \frac{\sqrt{2}}{2} and -\frac{\sqrt{2}}{2}. Substitute \frac{\sqrt{2}}{2} for y in the equation x=y to find the corresponding solution for x that satisfies both equations.
x=-\frac{\sqrt{2}}{2}
Now substitute -\frac{\sqrt{2}}{2} for y in the equation x=y and solve to find the corresponding solution for x that satisfies both equations.
x=\frac{\sqrt{2}}{2},y=\frac{\sqrt{2}}{2}\text{ or }x=-\frac{\sqrt{2}}{2},y=-\frac{\sqrt{2}}{2}
The system is now solved.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}