Solve {l}{x=ky+b}{9x^2+y^2=m^2}

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

Solve for x, y

$$x=\frac{-k\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+b}{9k^{2}+1}\text{, }y=-\frac{\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+9bk}{9k^{2}+1}$$

$$x=\frac{k\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+b}{9k^{2}+1}\text{, }y=\frac{\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}-9bk}{9k^{2}+1}\text{, }|b|<\frac{|m|}{3}\text{ or }\left(m=0\text{ and }b=0\right)\text{ or }\left(m\neq 0\text{ and }|k|\geq \frac{\sqrt{9b^{2}-m^{2}}}{3|m|}\text{ and }|b|\geq \frac{|m|}{3}\right)$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Subtract ky from both sides.

$$x-ky=b$$

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.

$$x+\left(-k\right)y=b,y^{2}+9x^{2}=m^{2}$$

Solve x+\left(-k\right)y=b for x by isolating x on the left hand side of the equal sign.

$$x+\left(-k\right)y=b$$

Subtract \left(-k\right)y from both sides of the equation.

$$x=ky+b$$

Substitute ky+b for x in the other equation, y^{2}+9x^{2}=m^{2}.

$$y^{2}+9\left(ky+b\right)^{2}=m^{2}$$

Square ky+b.

$$y^{2}+9\left(k^{2}y^{2}+2bky+b^{2}\right)=m^{2}$$

Multiply 9 times k^{2}y^{2}+2bky+b^{2}.

$$y^{2}+9k^{2}y^{2}+18bky+9b^{2}=m^{2}$$

Add y^{2} to 9k^{2}y^{2}.

$$\left(9k^{2}+1\right)y^{2}+18bky+9b^{2}=m^{2}$$

Subtract m^{2} from both sides of the equation.

$$\left(9k^{2}+1\right)y^{2}+18bky+9b^{2}-m^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+9k^{2} for a, 9\times 2kb for b, and 9b^{2}-m^{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-18bk±\sqrt{\left(18bk\right)^{2}-4\left(9k^{2}+1\right)\left(9b^{2}-m^{2}\right)}}{2\left(9k^{2}+1\right)}$$

Square 9\times 2kb.

$$y=\frac{-18bk±\sqrt{324b^{2}k^{2}-4\left(9k^{2}+1\right)\left(9b^{2}-m^{2}\right)}}{2\left(9k^{2}+1\right)}$$

Add 324k^{2}b^{2} to -4\left(1+9k^{2}\right)\left(9b^{2}-m^{2}\right).

$$y=\frac{-18bk±\sqrt{36k^{2}m^{2}+4m^{2}-36b^{2}}}{2\left(9k^{2}+1\right)}$$

Take the square root of 36k^{2}m^{2}+4m^{2}-36b^{2}.

$$y=\frac{-18bk±2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}}{2\left(9k^{2}+1\right)}$$

Multiply 2 times 1+9k^{2}.

$$y=\frac{-18bk±2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}}{18k^{2}+2}$$

Now solve the equation y=\frac{-18bk±2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}}{18k^{2}+2} when ± is plus. Add -18kb to 2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}.

$$y=\frac{2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-18bk}{18k^{2}+2}$$

Divide -18bk+2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}} by 2+18k^{2}.

$$y=\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-9bk}{9k^{2}+1}$$

Now solve the equation y=\frac{-18bk±2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}}{18k^{2}+2} when ± is minus. Subtract 2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}} from -18kb.

$$y=\frac{-2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-18bk}{18k^{2}+2}$$

Divide -18kb-2\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}} by 2+18k^{2}.

$$y=-\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}+9bk}{9k^{2}+1}$$

There are two solutions for y: \frac{-9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1} and -\frac{9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1}. Substitute \frac{-9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1} for y in the equation x=ky+b to find the corresponding solution for x that satisfies both equations.

$$x=k\times \frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-9bk}{9k^{2}+1}+b$$

Multiply k times \frac{-9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1}.

$$x=\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-9bk}{9k^{2}+1}k+b$$

Now substitute -\frac{9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1} for y in the equation x=ky+b and solve to find the corresponding solution for x that satisfies both equations.

$$x=k\left(-\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}+9bk}{9k^{2}+1}\right)+b$$

Multiply k times -\frac{9bk+\sqrt{-9b^{2}+m^{2}+9k^{2}m^{2}}}{9k^{2}+1}.

$$x=\left(-\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}+9bk}{9k^{2}+1}\right)k+b$$

The system is now solved.

$$x=\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-9bk}{9k^{2}+1}k+b,y=\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}-9bk}{9k^{2}+1}\text{ or }x=\left(-\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}+9bk}{9k^{2}+1}\right)k+b,y=-\frac{\sqrt{9k^{2}m^{2}+m^{2}-9b^{2}}+9bk}{9k^{2}+1}$$

Solve for x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{-k\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+b}{9k^{2}+1}\text{, }y=-\frac{\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+9bk}{9k^{2}+1}\text{; }x=\frac{k\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}+b}{9k^{2}+1}\text{, }y=\frac{\sqrt{m^{2}+9\left(km\right)^{2}-9b^{2}}-9bk}{9k^{2}+1}\text{, }&k\neq -\frac{1}{3}i\text{ and }k\neq \frac{1}{3}i\\x=\frac{m^{2}+9b^{2}}{18b}\text{, }y=\frac{m^{2}-9b^{2}}{18bk}\text{, }&b\neq 0\text{ and }\left(k=-\frac{1}{3}i\text{ or }k=\frac{1}{3}i\right)\\x=ky+b\text{, }y\in \mathrm{C}\text{, }&\left(k=-\frac{1}{3}i\text{ or }k=\frac{1}{3}i\right)\text{ and }b=0\text{ and }m=0\end{matrix}\right.$$