Solve {l}{x^2+y^2=9}{2y=x+1}

$$\left. \begin{array} { l } { x ^ { 2 } + y ^ { 2 } = 9 } \\ { 2 y = x + 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{-4\sqrt{11}-1}{5}\approx -2.853299832\text{, }y=\frac{2-2\sqrt{11}}{5}\approx -0.926649916$$

$$x=\frac{4\sqrt{11}-1}{5}\approx 2.453299832\text{, }y=\frac{2\sqrt{11}+2}{5}\approx 1.726649916$$

Steps Using Substitution and Quadratic Formula

Consider the second equation. Subtract x from both sides.

$$2y-x=1$$

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.

$$2y-x=1,x^{2}+y^{2}=9$$

Solve 2y-x=1 for y by isolating y on the left hand side of the equal sign.

$$2y-x=1$$

Subtract -x from both sides of the equation.

$$2y=x+1$$

Divide both sides by 2.

$$y=\frac{1}{2}x+\frac{1}{2}$$

Substitute \frac{1}{2}x+\frac{1}{2} for y in the other equation, x^{2}+y^{2}=9.

$$x^{2}+\left(\frac{1}{2}x+\frac{1}{2}\right)^{2}=9$$

Square \frac{1}{2}x+\frac{1}{2}.

$$x^{2}+\frac{1}{4}x^{2}+\frac{1}{2}x+\frac{1}{4}=9$$

Add x^{2} to \frac{1}{4}x^{2}.

$$\frac{5}{4}x^{2}+\frac{1}{2}x+\frac{1}{4}=9$$

Subtract 9 from both sides of the equation.

$$\frac{5}{4}x^{2}+\frac{1}{2}x-\frac{35}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\times \left(\frac{1}{2}\right)^{2} for a, 1\times \frac{1}{2}\times \frac{1}{2}\times 2 for b, and -\frac{35}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\times \frac{5}{4}\left(-\frac{35}{4}\right)}}{2\times \frac{5}{4}}$$

Square 1\times \frac{1}{2}\times \frac{1}{2}\times 2.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\times \frac{5}{4}\left(-\frac{35}{4}\right)}}{2\times \frac{5}{4}}$$

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

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-5\left(-\frac{35}{4}\right)}}{2\times \frac{5}{4}}$$

Multiply -5 times -\frac{35}{4}.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1+175}{4}}}{2\times \frac{5}{4}}$$

Add \frac{1}{4} to \frac{175}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{2}±\sqrt{44}}{2\times \frac{5}{4}}$$

Take the square root of 44.

$$x=\frac{-\frac{1}{2}±2\sqrt{11}}{2\times \frac{5}{4}}$$

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

$$x=\frac{-\frac{1}{2}±2\sqrt{11}}{\frac{5}{2}}$$

Now solve the equation x=\frac{-\frac{1}{2}±2\sqrt{11}}{\frac{5}{2}} when ± is plus. Add -\frac{1}{2} to 2\sqrt{11}.

$$x=\frac{2\sqrt{11}-\frac{1}{2}}{\frac{5}{2}}$$

Divide -\frac{1}{2}+2\sqrt{11} by \frac{5}{2} by multiplying -\frac{1}{2}+2\sqrt{11} by the reciprocal of \frac{5}{2}.

$$x=\frac{4\sqrt{11}-1}{5}$$

Now solve the equation x=\frac{-\frac{1}{2}±2\sqrt{11}}{\frac{5}{2}} when ± is minus. Subtract 2\sqrt{11} from -\frac{1}{2}.

$$x=\frac{-2\sqrt{11}-\frac{1}{2}}{\frac{5}{2}}$$

Divide -\frac{1}{2}-2\sqrt{11} by \frac{5}{2} by multiplying -\frac{1}{2}-2\sqrt{11} by the reciprocal of \frac{5}{2}.

$$x=\frac{-4\sqrt{11}-1}{5}$$

There are two solutions for x: \frac{-1+4\sqrt{11}}{5} and \frac{-1-4\sqrt{11}}{5}. Substitute \frac{-1+4\sqrt{11}}{5} for x in the equation y=\frac{1}{2}x+\frac{1}{2} to find the corresponding solution for y that satisfies both equations.

$$y=\frac{1}{2}\times \frac{4\sqrt{11}-1}{5}+\frac{1}{2}$$

Multiply \frac{1}{2} times \frac{-1+4\sqrt{11}}{5}.

$$y=\frac{4\sqrt{11}-1}{2\times 5}+\frac{1}{2}$$

Now substitute \frac{-1-4\sqrt{11}}{5} for x in the equation y=\frac{1}{2}x+\frac{1}{2} and solve to find the corresponding solution for y that satisfies both equations.

$$y=\frac{1}{2}\times \frac{-4\sqrt{11}-1}{5}+\frac{1}{2}$$

Multiply \frac{1}{2} times \frac{-1-4\sqrt{11}}{5}.

$$y=\frac{-4\sqrt{11}-1}{2\times 5}+\frac{1}{2}$$

The system is now solved.

$$y=\frac{4\sqrt{11}-1}{2\times 5}+\frac{1}{2},x=\frac{4\sqrt{11}-1}{5}\text{ or }y=\frac{-4\sqrt{11}-1}{2\times 5}+\frac{1}{2},x=\frac{-4\sqrt{11}-1}{5}$$