Solve {l}{y=kx-4k}{x^2/4+y^2/3=1}

$$\left\{ \begin{array} { l } { y = k x - 4 k } \\ { \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1 } \end{array} \right.$$

Solve for x, y

$$x=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}\text{, }y=-\frac{6k\left(\sqrt{1-4k^{2}}+2\right)}{4k^{2}+3}$$

$$x=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}\text{, }y=\frac{6k\left(\sqrt{1-4k^{2}}-2\right)}{4k^{2}+3}\text{, }|k|\leq \frac{1}{2}$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Subtract kx from both sides.

$$y-kx=-4k$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3x^{2}+4y^{2}=12$$

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.

$$y+\left(-k\right)x=-4k,3x^{2}+4y^{2}=12$$

Solve y+\left(-k\right)x=-4k for y by isolating y on the left hand side of the equal sign.

$$y+\left(-k\right)x=-4k$$

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

$$y=kx-4k$$

Substitute kx-4k for y in the other equation, 3x^{2}+4y^{2}=12.

$$3x^{2}+4\left(kx-4k\right)^{2}=12$$

Square kx-4k.

$$3x^{2}+4\left(k^{2}x^{2}+2\left(-4k\right)kx+\left(-4k\right)^{2}\right)=12$$

Multiply 4 times k^{2}x^{2}+2\left(-4k\right)kx+\left(-4k\right)^{2}.

$$3x^{2}+4k^{2}x^{2}+8\left(-4k\right)kx+4\left(-4k\right)^{2}=12$$

Add 3x^{2} to 4k^{2}x^{2}.

$$\left(4k^{2}+3\right)x^{2}+8\left(-4k\right)kx+4\left(-4k\right)^{2}=12$$

Subtract 12 from both sides of the equation.

$$\left(4k^{2}+3\right)x^{2}+8\left(-4k\right)kx+4\left(-4k\right)^{2}-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+4k^{2} for a, 4\times 2k\left(-4k\right) for b, and 64k^{2}-12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8\left(-4k\right)k±\sqrt{\left(8\left(-4k\right)k\right)^{2}-4\left(4k^{2}+3\right)\left(64k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Square 4\times 2k\left(-4k\right).

$$x=\frac{-8\left(-4k\right)k±\sqrt{1024k^{4}-4\left(4k^{2}+3\right)\left(64k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Multiply -4 times 3+4k^{2}.

$$x=\frac{-8\left(-4k\right)k±\sqrt{1024k^{4}+\left(-16k^{2}-12\right)\left(64k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Multiply -12-16k^{2} times 64k^{2}-12.

$$x=\frac{-8\left(-4k\right)k±\sqrt{1024k^{4}+144-576k^{2}-1024k^{4}}}{2\left(4k^{2}+3\right)}$$

Add 1024k^{4} to -576k^{2}-1024k^{4}+144.

$$x=\frac{-8\left(-4k\right)k±\sqrt{144-576k^{2}}}{2\left(4k^{2}+3\right)}$$

Take the square root of -576k^{2}+144.

$$x=\frac{-8\left(-4k\right)k±12\sqrt{1-4k^{2}}}{2\left(4k^{2}+3\right)}$$

Multiply 2 times 3+4k^{2}.

$$x=\frac{32k^{2}±12\sqrt{1-4k^{2}}}{8k^{2}+6}$$

Now solve the equation x=\frac{32k^{2}±12\sqrt{1-4k^{2}}}{8k^{2}+6} when ± is plus. Add 32k^{2} to 12\sqrt{1-4k^{2}}.

$$x=\frac{32k^{2}+12\sqrt{1-4k^{2}}}{8k^{2}+6}$$

Divide 32k^{2}+12\sqrt{1-4k^{2}} by 6+8k^{2}.

$$x=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}$$

Now solve the equation x=\frac{32k^{2}±12\sqrt{1-4k^{2}}}{8k^{2}+6} when ± is minus. Subtract 12\sqrt{1-4k^{2}} from 32k^{2}.

$$x=\frac{32k^{2}-12\sqrt{1-4k^{2}}}{8k^{2}+6}$$

Divide 32k^{2}-12\sqrt{1-4k^{2}} by 6+8k^{2}.

$$x=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}$$

There are two solutions for x: \frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{3+4k^{2}} and \frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{3+4k^{2}}. Substitute \frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{3+4k^{2}} for x in the equation y=kx-4k to find the corresponding solution for y that satisfies both equations.

$$y=k\times \frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}-4k$$

Multiply k times \frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{3+4k^{2}}.

$$y=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}k-4k$$

Now substitute \frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{3+4k^{2}} for x in the equation y=kx-4k and solve to find the corresponding solution for y that satisfies both equations.

$$y=k\times \frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}-4k$$

Multiply k times \frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{3+4k^{2}}.

$$y=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}k-4k$$

The system is now solved.

$$y=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}k-4k,x=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}\text{ or }y=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}k-4k,x=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}$$

Solve for x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{2\left(8k^{2}-3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}\text{, }y=-\frac{6k\left(\sqrt{1-4k^{2}}+2\right)}{4k^{2}+3}\text{; }x=\frac{2\left(8k^{2}+3\sqrt{1-4k^{2}}\right)}{4k^{2}+3}\text{, }y=\frac{6k\left(\sqrt{1-4k^{2}}-2\right)}{4k^{2}+3}\text{, }&k\neq -\frac{\sqrt{3}i}{2}\text{ and }k\neq \frac{\sqrt{3}i}{2}\\x=\frac{16k^{2}-3}{8k^{2}}\text{, }y=\frac{-16k^{2}-3}{8k}\text{, }&k=-\frac{\sqrt{3}i}{2}\text{ or }k=\frac{\sqrt{3}i}{2}\end{matrix}\right.$$