Solve the following​ equation, and check your solutions.

Question

StartFraction 11 Over y EndFraction plus 9 equals StartFraction 3 Over y EndFraction

Answer 1

To solve the equation $\frac{11}{y}+9=\frac{3}{y}$, let's first eliminate the fractions by multiplying both sides by $y$ (assuming $y\ne 0$):

$$y(\frac{11}{y}+9)=y\left(\frac{3}{y}\right)$$

This simplifies to:

$$11+9y=3$$

Next, we will isolate $y$ by rearranging the equation:

$$9y=3-11$$

$$9y=-8$$

Now, divide both sides by 9:

$$y=-\frac{8}{9}$$

Now, we have a potential solution. Let’s check this by substituting $y=-\frac{8}{9}$ back into the original equation:

1. Calculate $\frac{11}{y}$:

$$\frac{11}{-\frac{8}{9}}=11\cdot -\frac{9}{8}=-\frac{99}{8}$$

2. Calculate $\frac{3}{y}$:

$$\frac{3}{-\frac{8}{9}}=3\cdot -\frac{9}{8}=-\frac{27}{8}$$

3. Substitute the values into the original equation and simplify:

$$-\frac{99}{8}+9\stackrel{?}{=}-\frac{27}{8}$$

To combine $-\frac{99}{8}+9$, convert 9 to a fraction with a denominator of 8:

$$9=\frac{72}{8}$$

Now add:

$$-\frac{99}{8}+\frac{72}{8}=\frac{-99+72}{8}=\frac{-27}{8}$$

Thus:

$$-\frac{27}{8}=-\frac{27}{8}$$

The left side equals the right side, confirming that our solution is correct.

Therefore, the solution to the equation is:

$$\boxed{{\displaystyle -\frac{8}{9}}}$$