Solve 3t-1/3t+1=2-t-3/t+3

$$\frac{ 3t-1 }{ 3t+1 } = 2- \frac{ t-3 }{ t+3 }$$

Solve for t

$$t=-\frac{3}{5}=-0.6$$

Steps for Solving Linear Equation

Variable t cannot be equal to any of the values -3,-\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(t+3\right)\left(3t+1\right), the least common multiple of 3t+1,t+3.

$$\left(t+3\right)\left(3t-1\right)=\left(t+3\right)\left(3t+1\right)\times 2-\left(3t+1\right)\left(t-3\right)$$

Use the distributive property to multiply t+3 by 3t-1 and combine like terms.

$$3t^{2}+8t-3=\left(t+3\right)\left(3t+1\right)\times 2-\left(3t+1\right)\left(t-3\right)$$

Use the distributive property to multiply t+3 by 3t+1 and combine like terms.

$$3t^{2}+8t-3=\left(3t^{2}+10t+3\right)\times 2-\left(3t+1\right)\left(t-3\right)$$

Use the distributive property to multiply 3t^{2}+10t+3 by 2.

$$3t^{2}+8t-3=6t^{2}+20t+6-\left(3t+1\right)\left(t-3\right)$$

Use the distributive property to multiply 3t+1 by t-3 and combine like terms.

$$3t^{2}+8t-3=6t^{2}+20t+6-\left(3t^{2}-8t-3\right)$$

To find the opposite of 3t^{2}-8t-3, find the opposite of each term.

$$3t^{2}+8t-3=6t^{2}+20t+6-3t^{2}+8t+3$$

Combine 6t^{2} and -3t^{2} to get 3t^{2}.

$$3t^{2}+8t-3=3t^{2}+20t+6+8t+3$$

Combine 20t and 8t to get 28t.

$$3t^{2}+8t-3=3t^{2}+28t+6+3$$

Add 6 and 3 to get 9.

$$3t^{2}+8t-3=3t^{2}+28t+9$$

Subtract 3t^{2} from both sides.

$$3t^{2}+8t-3-3t^{2}=28t+9$$

Combine 3t^{2} and -3t^{2} to get 0.

$$8t-3=28t+9$$

Subtract 28t from both sides.

$$8t-3-28t=9$$

Combine 8t and -28t to get -20t.

$$-20t-3=9$$

Add 3 to both sides.

$$-20t=9+3$$

Add 9 and 3 to get 12.

$$-20t=12$$

Divide both sides by -20.

$$t=\frac{12}{-20}$$

Reduce the fraction \frac{12}{-20} to lowest terms by extracting and canceling out 4.

$$t=-\frac{3}{5}$$