Solve the following equation.Check your result in each case
`(9x+7)/2-(x-(x-2)/7)=36`

Answer

To solve the equation \((9x + 7)/2 - (x - (x - 2)/7) = 36\), we will follow these steps:

### Step 1: Simplify the Equation
Start with the original equation:
\[ \frac{9x + 7}{2} - \left(x - \frac{x - 2}{7}\right) = 36 \]

### Step 2: Simplify the Expression Inside the Parentheses
First, simplify the term \(x - \frac{x - 2}{7}\):
- Find a common denominator for the terms inside the parentheses. The common denominator is 7.
- Rewrite \(x\) as \(\frac{7x}{7}\):
\[ x - \frac{x - 2}{7} = \frac{7x}{7} - \frac{x - 2}{7} = \frac{7x - (x - 2)}{7} = \frac{7x - x + 2}{7} = \frac{6x + 2}{7} \]

### Step 3: Substitute Back into the Equation
Now substitute this back into the equation:
\[ \frac{9x + 7}{2} - \frac{6x + 2}{7} = 36 \]

### Step 4: Eliminate the Fractions
To eliminate the fractions, find the least common multiple (LCM) of the denominators (2 and 7), which is 14. Multiply the entire equation by 14:
\[ 14 \left(\frac{9x + 7}{2}\right) - 14 \left(\frac{6x + 2}{7}\right) = 14 \times 36 \]
This simplifies to:
\[ 7(9x + 7) - 2(6x + 2) = 504 \]

### Step 5: Distribute and Combine Like Terms
Distributing gives:
\[ 63x + 49 - 12x - 4 = 504 \]
Combine like terms:
\[ (63x - 12x) + (49 - 4) = 504 \]
This simplifies to:
\[ 51x + 45 = 504 \]

### Step 6: Isolate the Variable
Subtract 45 from both sides:
\[ 51x = 504 - 45 \]
\[ 51x = 459 \]

### Step 7: Solve for \(x\)
Divide both sides by 51:
\[ x = \frac{459}{51} \]
Calculating this gives:
\[ x = 9 \]

### Step 8: Check the Solution
Substitute \(x = 9\) back into the original equation to verify:
\[ \frac{9(9) + 7}{2} - \left(9 - \frac{9 - 2}{7}\right) = 36 \]
Calculating:
\[ \frac{81 + 7}{2} - \left(9 - \frac{7}{7}\right) = 36 \]
This simplifies to:
\[ \frac{88}{2} - (9 - 1) = 36 \]
\[ 44 - 8 = 36 \]
Since both sides are equal, our solution is verified.

### Final Answer:
\[ x = 9 \]

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