Solve the equations and check your answer in the case:
`(8-3x)/(5x+31)=(2)/(3)`

Answer

To solve the equation \(\frac{8-3x}{5x+31}=\frac{2}{3}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:
\[ 3(8 - 3x) = 2(5x + 31) \]

### Step 2: Distribute
Next, we distribute on both sides:
\[ 24 - 9x = 10x + 62 \]

### Step 3: Move all terms involving \(x\) to one side
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 24 - 62 = 10x + 9x \]
This simplifies to:
\[ -38 = 19x \]

### Step 4: Solve for \(x\)
Now, we divide both sides by 19 to solve for \(x\):
\[ x = \frac{-38}{19} = -2 \]

### Step 5: Check the solution
To verify our solution, we substitute \(x = -2\) back into the original equation:
\[ \frac{8 - 3(-2)}{5(-2) + 31} \]

Calculating the numerator:
\[ 8 + 6 = 14 \]

Calculating the denominator:
\[ -10 + 31 = 21 \]

Thus, we have:
\[ \frac{14}{21} = \frac{2}{3} \]

Since both sides of the equation are equal, our solution is confirmed.

### Final Answer
The solution to the equation is:
\[ x = -2 \]