Solve {l}{1/x+2=4/3}{text{Solvefor}ytext{where}}{y=1/x+1/x+1}

$$\left. \begin{array} { l } { \frac{1}{x + 2} = \frac{4}{3} }\\ { \text{Solve for } y \text{ where} } \\ { y = \frac{1}{x} + \frac{1}{x + 1} } \end{array} \right.$$

Solve for x, y

$$y = -\frac{24}{5} = -4\frac{4}{5} = -4.8$$

Solution Steps

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

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

Use the distributive property to multiply 4 by x+2.

$$3=4x+8$$

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

$$4x+8=3$$

Subtract 8 from both sides.

$$4x=3-8$$

Subtract 8 from 3 to get -5.

$$4x=-5$$

Divide both sides by 4.

$$x=-\frac{5}{4}$$

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

$$y=\frac{1}{-\frac{5}{4}}+\frac{1}{-\frac{5}{4}+1}$$

Divide 1 by -\frac{5}{4} by multiplying 1 by the reciprocal of -\frac{5}{4}.

$$y=1\left(-\frac{4}{5}\right)+\frac{1}{-\frac{5}{4}+1}$$

Multiply 1 and -\frac{4}{5} to get -\frac{4}{5}.

$$y=-\frac{4}{5}+\frac{1}{-\frac{5}{4}+1}$$

Add -\frac{5}{4} and 1 to get -\frac{1}{4}.

$$y=-\frac{4}{5}+\frac{1}{-\frac{1}{4}}$$

Divide 1 by -\frac{1}{4} by multiplying 1 by the reciprocal of -\frac{1}{4}.

$$y=-\frac{4}{5}+1\left(-4\right)$$

Multiply 1 and -4 to get -4.

$$y=-\frac{4}{5}-4$$

Subtract 4 from -\frac{4}{5} to get -\frac{24}{5}.

$$y=-\frac{24}{5}$$

The system is now solved.

$$x=-\frac{5}{4} y=-\frac{24}{5}$$