Solve {l}{x+3/x+1-2x+5/x+2=x-5/3x+3-4x-3/3x+6}{y=2x-9*3x+9*13x+69}{text{Solvefor}ztext{where}}{z=y}

$$\left. \begin{array} { l } { \frac{x + 3}{x + 1} - \frac{2 x + 5}{x + 2} = \frac{x - 5}{3 x + 3} - \frac{4 x - 3}{3 x + 6} }\\ { y = 2 x - 9 \cdot 3 x + 9 \cdot 13 x + 69 }\\ { \text{Solve for } z \text{ where} } \\ { z = y } \end{array} \right.$$

Solve for x, y, z

$$z=529$$

Solution Steps

Consider the first equation. Variable x cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,3x+3,3x+6.

$$\left(3x+6\right)\left(x+3\right)-\left(3x+3\right)\left(2x+5\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply 3x+6 by x+3 and combine like terms.

$$3x^{2}+15x+18-\left(3x+3\right)\left(2x+5\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply 3x+3 by 2x+5 and combine like terms.

$$3x^{2}+15x+18-\left(6x^{2}+21x+15\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

To find the opposite of 6x^{2}+21x+15, find the opposite of each term.

$$3x^{2}+15x+18-6x^{2}-21x-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

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

$$-3x^{2}+15x+18-21x-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Combine 15x and -21x to get -6x.

$$-3x^{2}-6x+18-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Subtract 15 from 18 to get 3.

$$-3x^{2}-6x+3=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply x+2 by x-5 and combine like terms.

$$-3x^{2}-6x+3=x^{2}-3x-10-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply x+1 by 4x-3 and combine like terms.

$$-3x^{2}-6x+3=x^{2}-3x-10-\left(4x^{2}+x-3\right)$$

To find the opposite of 4x^{2}+x-3, find the opposite of each term.

$$-3x^{2}-6x+3=x^{2}-3x-10-4x^{2}-x+3$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+3=-3x^{2}-3x-10-x+3$$

Combine -3x and -x to get -4x.

$$-3x^{2}-6x+3=-3x^{2}-4x-10+3$$

Add -10 and 3 to get -7.

$$-3x^{2}-6x+3=-3x^{2}-4x-7$$

Add 3x^{2} to both sides.

$$-3x^{2}-6x+3+3x^{2}=-4x-7$$

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

$$-6x+3=-4x-7$$

Add 4x to both sides.

$$-6x+3+4x=-7$$

Combine -6x and 4x to get -2x.

$$-2x+3=-7$$

Subtract 3 from both sides.

$$-2x=-7-3$$

Subtract 3 from -7 to get -10.

$$-2x=-10$$

Divide both sides by -2.

$$x=\frac{-10}{-2}$$

Divide -10 by -2 to get 5.

$$x=5$$

Consider the second equation. Insert the known values of variables into the equation.

$$y=2\times 5-9\times 3\times 5+9\times 13\times 5+69$$

Multiply 2 and 5 to get 10.

$$y=10-9\times 3\times 5+9\times 13\times 5+69$$

Multiply 9 and 3 to get 27.

$$y=10-27\times 5+9\times 13\times 5+69$$

Multiply 27 and 5 to get 135.

$$y=10-135+9\times 13\times 5+69$$

Subtract 135 from 10 to get -125.

$$y=-125+9\times 13\times 5+69$$

Multiply 9 and 13 to get 117.

$$y=-125+117\times 5+69$$

Multiply 117 and 5 to get 585.

$$y=-125+585+69$$

Add -125 and 585 to get 460.

$$y=460+69$$

Add 460 and 69 to get 529.

$$y=529$$

Consider the third equation. Insert the known values of variables into the equation.

$$z=529$$

The system is now solved.

$$x=5 y=529 z=529$$