Solve 2x-3/x-1=4x-1/2x+3

$$\frac{ 2x-3 }{ x-1 } = \frac{ 4x-1 }{ 2x+3 }$$

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -\frac{3}{2},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(2x+3\right), the least common multiple of x-1,2x+3.

$$\left(2x+3\right)\left(2x-3\right)=\left(x-1\right)\left(4x-1\right)$$

Consider \left(2x+3\right)\left(2x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$\left(2x\right)^{2}-9=\left(x-1\right)\left(4x-1\right)$$

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}-9=\left(x-1\right)\left(4x-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-9=\left(x-1\right)\left(4x-1\right)$$

Use the distributive property to multiply x-1 by 4x-1 and combine like terms.

$$4x^{2}-9=4x^{2}-5x+1$$

Subtract 4x^{2} from both sides.

$$4x^{2}-9-4x^{2}=-5x+1$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-9=-5x+1$$

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

$$-5x+1=-9$$

Subtract 1 from both sides.

$$-5x=-9-1$$

Subtract 1 from -9 to get -10.

$$-5x=-10$$

Divide both sides by -5.

$$x=\frac{-10}{-5}$$

Divide -10 by -5 to get 2.

$$x=2$$