Solve {c}{y-x=1}{x^2+y^2=5}

$$\left\{ \begin{array} { c } { y - x = 1 } \\ { x ^ { 2 } + y ^ { 2 } = 5 } \end{array} \right.$$

Solve for y, x

$$x=1\text{, }y=2$$

$$x=-2\text{, }y=-1$$

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.

$$y-x=1,x^{2}+y^{2}=5$$

Solve y-x=1 for y by isolating y on the left hand side of the equal sign.

$$y-x=1$$

Subtract -x from both sides of the equation.

$$y=x+1$$

Substitute x+1 for y in the other equation, x^{2}+y^{2}=5.

$$x^{2}+\left(x+1\right)^{2}=5$$

Square x+1.

$$x^{2}+x^{2}+2x+1=5$$

Add x^{2} to x^{2}.

$$2x^{2}+2x+1=5$$

Subtract 5 from both sides of the equation.

$$2x^{2}+2x-4=0$$

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

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

Square 1\times 1\times 1\times 2.

$$x=\frac{-2±\sqrt{4-4\times 2\left(-4\right)}}{2\times 2}$$

Multiply -4 times 1+1\times 1^{2}.

$$x=\frac{-2±\sqrt{4-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$x=\frac{-2±\sqrt{4+32}}{2\times 2}$$

Add 4 to 32.

$$x=\frac{-2±\sqrt{36}}{2\times 2}$$

Take the square root of 36.

$$x=\frac{-2±6}{2\times 2}$$

Multiply 2 times 1+1\times 1^{2}.

$$x=\frac{-2±6}{4}$$

Now solve the equation x=\frac{-2±6}{4} when ± is plus. Add -2 to 6.

$$x=\frac{4}{4}$$

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-2±6}{4} when ± is minus. Subtract 6 from -2.

$$x=-\frac{8}{4}$$

Divide -8 by 4.

$$x=-2$$

There are two solutions for x: 1 and -2. Substitute 1 for x in the equation y=x+1 to find the corresponding solution for y that satisfies both equations.

$$y=1+1$$

Add 1\times 1 to 1.

$$y=2$$

Now substitute -2 for x in the equation y=x+1 and solve to find the corresponding solution for y that satisfies both equations.

$$y=-2+1$$

Add -2 to 1.

$$y=-1$$

The system is now solved.

$$y=2,x=1\text{ or }y=-1,x=-2$$