Solve 2(m^2+1/m^2)-3(x-1/m)-1=0

$$2 ( m ^ { 2 } + \frac { 1 } { m ^ { 2 } } ) - 3 ( x - \frac { 1 } { m } ) - 1 = 0$$

Solve for x

$$x=\frac{2m^{2}}{3}-\frac{1}{3}+\frac{1}{m}+\frac{2}{3m^{2}}$$

$$m\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by m^{2}, the least common multiple of m^{2},m.

$$2\left(m^{2}+\frac{1}{m^{2}}\right)m^{2}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply m^{2} times \frac{m^{2}}{m^{2}}.

$$2\left(\frac{m^{2}m^{2}}{m^{2}}+\frac{1}{m^{2}}\right)m^{2}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Since \frac{m^{2}m^{2}}{m^{2}} and \frac{1}{m^{2}} have the same denominator, add them by adding their numerators.

$$2\times \frac{m^{2}m^{2}+1}{m^{2}}m^{2}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Do the multiplications in m^{2}m^{2}+1.

$$2\times \frac{m^{4}+1}{m^{2}}m^{2}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Express 2\times \frac{m^{4}+1}{m^{2}} as a single fraction.

$$\frac{2\left(m^{4}+1\right)}{m^{2}}m^{2}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Express \frac{2\left(m^{4}+1\right)}{m^{2}}m^{2} as a single fraction.

$$\frac{2\left(m^{4}+1\right)m^{2}}{m^{2}}-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Cancel out m^{2} in both numerator and denominator.

$$2\left(m^{4}+1\right)-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Use the distributive property to multiply 2 by m^{4}+1.

$$2m^{4}+2-3\left(x-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{m}{m}.

$$2m^{4}+2-3\left(\frac{xm}{m}-\frac{1}{m}\right)m^{2}+m^{2}\left(-1\right)=0$$

Since \frac{xm}{m} and \frac{1}{m} have the same denominator, subtract them by subtracting their numerators.

$$2m^{4}+2-3\times \frac{xm-1}{m}m^{2}+m^{2}\left(-1\right)=0$$

Express -3\times \frac{xm-1}{m} as a single fraction.

$$2m^{4}+2+\frac{-3\left(xm-1\right)}{m}m^{2}+m^{2}\left(-1\right)=0$$

Express \frac{-3\left(xm-1\right)}{m}m^{2} as a single fraction.

$$2m^{4}+2+\frac{-3\left(xm-1\right)m^{2}}{m}+m^{2}\left(-1\right)=0$$

Cancel out m in both numerator and denominator.

$$2m^{4}+2-3m\left(mx-1\right)+m^{2}\left(-1\right)=0$$

Use the distributive property to multiply -3m by mx-1.

$$2m^{4}+2-3xm^{2}+3m+m^{2}\left(-1\right)=0$$

Subtract 2m^{4} from both sides. Anything subtracted from zero gives its negation.

$$2-3xm^{2}+3m+m^{2}\left(-1\right)=-2m^{4}$$

Subtract 2 from both sides.

$$-3xm^{2}+3m+m^{2}\left(-1\right)=-2m^{4}-2$$

Subtract 3m from both sides.

$$-3xm^{2}+m^{2}\left(-1\right)=-2m^{4}-2-3m$$

Subtract m^{2}\left(-1\right) from both sides.

$$-3xm^{2}=-2m^{4}-2-3m-m^{2}\left(-1\right)$$

Multiply -1 and -1 to get 1.

$$-3xm^{2}=-2m^{4}-2-3m+m^{2}$$

The equation is in standard form.

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

Divide both sides by -3m^{2}.

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

Dividing by -3m^{2} undoes the multiplication by -3m^{2}.

$$x=\frac{-2m^{4}+m^{2}-3m-2}{-3m^{2}}$$

Divide -2m^{4}+m^{2}-3m-2 by -3m^{2}.

$$x=\frac{2m^{2}}{3}-\frac{1}{3}+\frac{m+\frac{2}{3}}{m^{2}}$$