Two adjacent angles on a straight line are `(5x-6)^(@) and 7(x+6)^(@)` . Find the value of x and magenitude of both the angles.

Answer

To solve the problem, we need to find the value of \( x \) and the magnitudes of the two adjacent angles formed on a straight line. The angles are given as \( (5x - 6)^\circ \) and \( (7(x + 6))^\circ \).

### Step-by-Step Solution:

1. **Set up the equation for adjacent angles on a straight line**:
Since the angles are adjacent and form a straight line, their sum is equal to \( 180^\circ \).
\[ (5x - 6) + (7(x + 6)) = 180 \]

2. **Expand the equation**:
First, we will expand the second angle:
\[ 7(x + 6) = 7x + 42 \]
Therefore, the equation becomes:
\[ (5x - 6) + (7x + 42) = 180 \]

3. **Combine like terms**:
Combine the \( x \) terms and the constant terms:
\[ 5x + 7x - 6 + 42 = 180 \]
This simplifies to:
\[ 12x + 36 = 180 \]

4. **Isolate the variable \( x \)**:
Subtract \( 36 \) from both sides:
\[ 12x = 180 - 36 \]
\[ 12x = 144 \]

5. **Solve for \( x \)**:
Divide both sides by \( 12 \):
\[ x = \frac{144}{12} \]
\[ x = 12 \]

6. **Find the magnitudes of both angles**:
Now that we have \( x = 12 \), we can find the magnitudes of the angles.

- For the first angle:
\[ 5x - 6 = 5(12) - 6 = 60 - 6 = 54^\circ \]

- For the second angle:
\[ 7(x + 6) = 7(12 + 6) = 7(18) = 126^\circ \]

### Summary of Results:
- The value of \( x \) is \( 12 \).
- The magnitudes of the angles are \( 54^\circ \) and \( 126^\circ \).