Solve 1/2n^3=n^2+3n-10/n^3-n+5/n^2

$$\frac { 1 } { 2 n ^ { 3 } } = \frac { n ^ { 2 } + 3 n - 10 } { n ^ { 3 } } - \frac { n + 5 } { n ^ { 2 } }$$

Solve for n

$$n = -\frac{21}{4} = -5\frac{1}{4} = -5.25$$

Steps for Solving Linear Equation

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2n^{3}, the least common multiple of 2n^{3},n^{3},n^{2}.

$$1=2\left(n^{2}+3n-10\right)-2n\left(n+5\right)$$

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

$$1=2n^{2}+6n-20-2n\left(n+5\right)$$

Use the distributive property to multiply 2n by n+5.

$$1=2n^{2}+6n-20-\left(2n^{2}+10n\right)$$

To find the opposite of 2n^{2}+10n, find the opposite of each term.

$$1=2n^{2}+6n-20-2n^{2}-10n$$

Combine 2n^{2} and -2n^{2} to get 0.

$$1=6n-20-10n$$

Combine 6n and -10n to get -4n.

$$1=-4n-20$$

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

$$-4n-20=1$$

Add 20 to both sides.

$$-4n=1+20$$

Add 1 and 20 to get 21.

$$-4n=21$$

Divide both sides by -4.

$$n=\frac{21}{-4}$$

Fraction \frac{21}{-4} can be rewritten as -\frac{21}{4} by extracting the negative sign.

$$n=-\frac{21}{4}$$