Solve for r (complex solution)
\left\{\begin{matrix}r=-\frac{-5x+t-4}{x}\text{, }&x\neq 0\text{ and }x\neq 2\text{ and }x\neq -5\\r\in \mathrm{C}\text{, }&x=0\text{ and }t=4\end{matrix}\right.
Solve for r
\left\{\begin{matrix}r=-\frac{-5x+t-4}{x}\text{, }&x\neq 0\text{ and }x\neq 2\text{ and }x\neq -5\\r\in \mathrm{R}\text{, }&x=0\text{ and }t=4\end{matrix}\right.
Solve for t
t=4+5x-rx
x\neq 2\text{ and }x\neq -5
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\left(x+5\right)\times 2+\left(x-2\right)\times 3=rx+t
Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5,\left(x-2\right)\left(x+5\right).
2x+10+\left(x-2\right)\times 3=rx+t
Use the distributive property to multiply x+5 by 2.
2x+10+3x-6=rx+t
Use the distributive property to multiply x-2 by 3.
5x+10-6=rx+t
Combine 2x and 3x to get 5x.
5x+4=rx+t
Subtract 6 from 10 to get 4.
rx+t=5x+4
Swap sides so that all variable terms are on the left hand side.
rx=5x+4-t
Subtract t from both sides.
xr=5x-t+4
The equation is in standard form.
\frac{xr}{x}=\frac{5x-t+4}{x}
Divide both sides by x.
r=\frac{5x-t+4}{x}
Dividing by x undoes the multiplication by x.
\left(x+5\right)\times 2+\left(x-2\right)\times 3=rx+t
Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5,\left(x-2\right)\left(x+5\right).
2x+10+\left(x-2\right)\times 3=rx+t
Use the distributive property to multiply x+5 by 2.
2x+10+3x-6=rx+t
Use the distributive property to multiply x-2 by 3.
5x+10-6=rx+t
Combine 2x and 3x to get 5x.
5x+4=rx+t
Subtract 6 from 10 to get 4.
rx+t=5x+4
Swap sides so that all variable terms are on the left hand side.
rx=5x+4-t
Subtract t from both sides.
xr=5x-t+4
The equation is in standard form.
\frac{xr}{x}=\frac{5x-t+4}{x}
Divide both sides by x.
r=\frac{5x-t+4}{x}
Dividing by x undoes the multiplication by x.
\left(x+5\right)\times 2+\left(x-2\right)\times 3=rx+t
Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5,\left(x-2\right)\left(x+5\right).
2x+10+\left(x-2\right)\times 3=rx+t
Use the distributive property to multiply x+5 by 2.
2x+10+3x-6=rx+t
Use the distributive property to multiply x-2 by 3.
5x+10-6=rx+t
Combine 2x and 3x to get 5x.
5x+4=rx+t
Subtract 6 from 10 to get 4.
rx+t=5x+4
Swap sides so that all variable terms are on the left hand side.
t=5x+4-rx
Subtract rx from both sides.
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