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

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

Solve for x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -\frac{3}{2},-\frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by \left(2x+3\right)\left(4x+1\right), the least common multiple of 4x+1,2x+3.

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

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.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

$$4x^{2}-9=4x^{2}-3x-1$$

Subtract 4x^{2} from both sides.

$$4x^{2}-9-4x^{2}=-3x-1$$

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

$$-9=-3x-1$$

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

$$-3x-1=-9$$

Add 1 to both sides.

$$-3x=-9+1$$

Add -9 and 1 to get -8.

$$-3x=-8$$

Divide both sides by -3.

$$x=\frac{-8}{-3}$$

Fraction \frac{-8}{-3} can be simplified to \frac{8}{3} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{8}{3}$$