Solve 4x+3y+2x+6y=500+520quadk:x+y=?

$$4 x + 3 y + 2 x + 6 y = 500 + 520 \quad k : x + y = ?$$

Solve for k

$$k=-\frac{x\left(250-4y-3x\right)}{260}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$4xx+3yx+2xx+6yx=x\times 500+520k+xy$$

Multiply x and x to get x^{2}.

$$4x^{2}+3yx+2xx+6yx=x\times 500+520k+xy$$

Multiply x and x to get x^{2}.

$$4x^{2}+3yx+2x^{2}+6yx=x\times 500+520k+xy$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}+3yx+6yx=x\times 500+520k+xy$$

Combine 3yx and 6yx to get 9yx.

$$6x^{2}+9yx=x\times 500+520k+xy$$

Swap sides so that all variable terms are on the left hand side.

$$x\times 500+520k+xy=6x^{2}+9yx$$

Subtract x\times 500 from both sides.

$$520k+xy=6x^{2}+9yx-x\times 500$$

Subtract xy from both sides.

$$520k=6x^{2}+9yx-x\times 500-xy$$

Multiply -1 and 500 to get -500.

$$520k=6x^{2}+9yx-500x-xy$$

Combine 9yx and -xy to get 8yx.

$$520k=6x^{2}+8yx-500x$$

The equation is in standard form.

$$520k=6x^{2}+8xy-500x$$

Divide both sides by 520.

$$\frac{520k}{520}=\frac{2x\left(3x+4y-250\right)}{520}$$

Dividing by 520 undoes the multiplication by 520.

$$k=\frac{2x\left(3x+4y-250\right)}{520}$$

Divide 2x\left(-250+3x+4y\right) by 520.

$$k=\frac{x\left(3x+4y-250\right)}{260}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{4y^{2}-500y+780k+15625}-2y+125}{3}\text{, }&\left(arg(125-2y)\geq \pi \text{ and }y\neq \frac{125}{2}\right)\text{ or }k\neq 0\\x=\frac{\sqrt{4y^{2}-500y+780k+15625}-2y+125}{3}\text{, }&\left(arg(2y-125)\geq \pi \text{ and }y\neq \frac{125}{2}\right)\text{ or }k\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{4y^{2}-500y+780k+15625}-2y+125}{3}\text{, }&\left(y\leq -\sqrt{-195k}+\frac{125}{2}\text{ or }y>\frac{125}{2}\text{ or }k>0\text{ or }y\geq \sqrt{-195k}+\frac{125}{2}\right)\text{ and }\left(y\leq -\sqrt{-195k}+\frac{125}{2}\text{ or }y\geq \sqrt{-195k}+\frac{125}{2}\text{ or }k\geq 0\right)\text{ and }k\geq -\frac{\left(2y-125\right)^{2}}{780}\text{ and }\left(k\neq 0\text{ or }y>\frac{125}{2}\right)\\x=\frac{\sqrt{4y^{2}-500y+780k+15625}-2y+125}{3}\text{, }&\left(y\leq -\sqrt{-195k}+\frac{125}{2}\text{ and }k\geq -\frac{\left(2y-125\right)^{2}}{780}\text{ and }k<0\right)\text{ or }\left(y\geq \sqrt{-195k}+\frac{125}{2}\text{ and }k\geq -\frac{\left(2y-125\right)^{2}}{780}\text{ and }k<0\right)\text{ or }\left(y<\frac{125}{2}\text{ and }y\leq -\sqrt{-195k}+\frac{125}{2}\text{ and }k\geq -\frac{\left(2y-125\right)^{2}}{780}\text{ and }k\leq 0\right)\text{ or }\left(k\geq 0\text{ and }y<\frac{125}{2}\right)\text{ or }k>0\end{matrix}\right.$$