Solve {l}{y=-3/4kx}{3x^2+4y^2=18}

$$\left. \begin{array} { l } { y = - \frac { 3 } { 4 k } x } \\ { 3 x ^ { 2 } + 4 y ^ { 2 } = 18 } \end{array} \right.$$

Solve for x, y

$$x=2\sqrt{\frac{6}{4k^{2}+3}}k\text{, }y=-\frac{3\sqrt{\frac{6}{4k^{2}+3}}}{2}$$

$$x=-2\sqrt{\frac{6}{4k^{2}+3}}k\text{, }y=\frac{3\sqrt{\frac{6}{4k^{2}+3}}}{2}\text{, }k\neq 0$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides of the equation by 4k.

$$y\times 4k=\left(-\frac{3}{4k}\right)x\times 4k$$

Express \left(-\frac{3}{4k}\right)x as a single fraction.

$$y\times 4k=\frac{-3x}{4k}\times 4k$$

Express \frac{-3x}{4k}\times 4 as a single fraction.

$$y\times 4k=\frac{-3x\times 4}{4k}k$$

Cancel out 4 in both numerator and denominator.

$$y\times 4k=\frac{-3x}{k}k$$

Express \frac{-3x}{k}k as a single fraction.

$$y\times 4k=\frac{-3xk}{k}$$

Cancel out k in both numerator and denominator.

$$y\times 4k=-3x$$

Add 3x to both sides.

$$y\times 4k+3x=0$$

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.

$$4ky+3x=0,3x^{2}+4y^{2}=18$$

Solve 4ky+3x=0 for y by isolating y on the left hand side of the equal sign.

$$4ky+3x=0$$

Subtract 3x from both sides of the equation.

$$4ky=-3x$$

Divide both sides by 4k.

$$y=\left(-\frac{3}{4k}\right)x$$

Substitute \left(-\frac{3}{4k}\right)x for y in the other equation, 3x^{2}+4y^{2}=18.

$$3x^{2}+4\left(\left(-\frac{3}{4k}\right)x\right)^{2}=18$$

Square \left(-\frac{3}{4k}\right)x.

$$3x^{2}+4\left(-\frac{3}{4k}\right)^{2}x^{2}=18$$

Add 3x^{2} to 4\left(-\frac{3}{4k}\right)^{2}x^{2}.

$$\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)x^{2}=18$$

Subtract 18 from both sides of the equation.

$$\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)x^{2}-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+4\left(-\frac{3}{4k}\right)^{2} for a, 4\times 0\times 2\left(-\frac{3}{4k}\right) for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)\left(-18\right)}}{2\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)}$$

Square 4\times 0\times 2\left(-\frac{3}{4k}\right).

$$x=\frac{0±\sqrt{-4\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)\left(-18\right)}}{2\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)}$$

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

$$x=\frac{0±\sqrt{\left(-12-\frac{9}{k^{2}}\right)\left(-18\right)}}{2\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)}$$

Multiply -12-\frac{9}{k^{2}} times -18.

$$x=\frac{0±\sqrt{216+\frac{162}{k^{2}}}}{2\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)}$$

Take the square root of 216+\frac{162}{k^{2}}.

$$x=\frac{0±\frac{3\sqrt{24k^{2}+18}}{|k|}}{2\left(3+4\left(-\frac{3}{4k}\right)^{2}\right)}$$

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

$$x=\frac{0±\frac{3\sqrt{24k^{2}+18}}{|k|}}{6+\frac{9}{2k^{2}}}$$

Now solve the equation x=\frac{0±\frac{3\sqrt{24k^{2}+18}}{|k|}}{6+\frac{9}{2k^{2}}} when ± is plus.

$$x=\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}$$

Now solve the equation x=\frac{0±\frac{3\sqrt{24k^{2}+18}}{|k|}}{6+\frac{9}{2k^{2}}} when ± is minus.

$$x=-\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}$$

There are two solutions for x: \frac{12k^{2}}{|k|\sqrt{24k^{2}+18}} and -\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}. Substitute \frac{12k^{2}}{|k|\sqrt{24k^{2}+18}} for x in the equation y=\left(-\frac{3}{4k}\right)x to find the corresponding solution for y that satisfies both equations.

$$y=\left(-\frac{3}{4k}\right)\times \frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}$$

Now substitute -\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}} for x in the equation y=\left(-\frac{3}{4k}\right)x and solve to find the corresponding solution for y that satisfies both equations.

$$y=\left(-\frac{3}{4k}\right)\left(-\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}\right)$$

The system is now solved.

$$y=\left(-\frac{3}{4k}\right)\times \frac{12k^{2}}{|k|\sqrt{24k^{2}+18}},x=\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}\text{ or }y=\left(-\frac{3}{4k}\right)\left(-\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}\right),x=-\frac{12k^{2}}{|k|\sqrt{24k^{2}+18}}$$

Solve for x, y (complex solution)

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

$$x=2\sqrt{6}\left(4k^{2}+3\right)^{-\frac{1}{2}}k\text{, }y=-\frac{3\sqrt{6}\left(4k^{2}+3\right)^{-\frac{1}{2}}}{2}\text{, }k\neq -\frac{\sqrt{3}i}{2}\text{ and }k\neq \frac{\sqrt{3}i}{2}\text{ and }k\neq 0$$