Solve {l}{5x+1/2=3x-1/3}{y={(5x+1)}{(2x+3)}}{text{Solvefor}ztext{where}}{z=y}

$$\left. \begin{array} { l } { \frac{5 x + 1}{2} = \frac{3 x - 1}{3} }\\ { y = {(5 x + 1)} {(2 x + 3)} }\\ { \text{Solve for } z \text{ where} } \\ { z = y } \end{array} \right.$$

Solve for x, y, z

$$z = -\frac{272}{81} = -3\frac{29}{81} \approx -3.358024691$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

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

Use the distributive property to multiply 3 by 5x+1.

$$15x+3=2\left(3x-1\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$15x+3=6x-2$$

Subtract 6x from both sides.

$$15x+3-6x=-2$$

Combine 15x and -6x to get 9x.

$$9x+3=-2$$

Subtract 3 from both sides.

$$9x=-2-3$$

Subtract 3 from -2 to get -5.

$$9x=-5$$

Divide both sides by 9.

$$x=-\frac{5}{9}$$

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

$$y=\left(5\left(-\frac{5}{9}\right)+1\right)\left(2\left(-\frac{5}{9}\right)+3\right)$$

Multiply 5 and -\frac{5}{9} to get -\frac{25}{9}.

$$y=\left(-\frac{25}{9}+1\right)\left(2\left(-\frac{5}{9}\right)+3\right)$$

Add -\frac{25}{9} and 1 to get -\frac{16}{9}.

$$y=-\frac{16}{9}\left(2\left(-\frac{5}{9}\right)+3\right)$$

Multiply 2 and -\frac{5}{9} to get -\frac{10}{9}.

$$y=-\frac{16}{9}\left(-\frac{10}{9}+3\right)$$

Add -\frac{10}{9} and 3 to get \frac{17}{9}.

$$y=-\frac{16}{9}\times \frac{17}{9}$$

Multiply -\frac{16}{9} and \frac{17}{9} to get -\frac{272}{81}.

$$y=-\frac{272}{81}$$

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

$$z=-\frac{272}{81}$$

The system is now solved.

$$x=-\frac{5}{9} y=-\frac{272}{81} z=-\frac{272}{81}$$