The point (2, 1), (8, 5) and (x, 7) lie on a straight line. Then the value of x is :

Options

A. 10
B. 11
C. 12
D. `(35)/(3)`

Answer

To find the value of \( x \) such that the points \( (2, 1) \), \( (8, 5) \), and \( (x, 7) \) lie on the same straight line, we can follow these steps:

### Step 1: Use the two-point form of the equation of a line
The two-point form of the equation of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

Here, we can take \( (x_1, y_1) = (2, 1) \) and \( (x_2, y_2) = (8, 5) \).

### Step 2: Substitute the values into the equation
Substituting the values into the equation:

\[ y - 1 = \frac{5 - 1}{8 - 2} (x - 2) \]

This simplifies to:

\[ y - 1 = \frac{4}{6} (x - 2) \]

### Step 3: Simplify the equation
Simplifying \( \frac{4}{6} \) gives \( \frac{2}{3} \):

\[ y - 1 = \frac{2}{3} (x - 2) \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation:

\[ y - 1 = \frac{2}{3}x - \frac{4}{3} \]

Multiplying through by 3 to eliminate the fraction:

\[ 3(y - 1) = 2x - 4 \]

This expands to:

\[ 3y - 3 = 2x - 4 \]

Rearranging gives:

\[ 3y - 2x + 1 = 0 \]

### Step 5: Substitute \( y = 7 \)
Now we substitute \( y = 7 \) into the equation:

\[ 3(7) - 2x + 1 = 0 \]

This simplifies to:

\[ 21 - 2x + 1 = 0 \]

### Step 6: Solve for \( x \)
Combining the constants gives:

\[ 22 - 2x = 0 \]

Thus, we have:

\[ 2x = 22 \]

Dividing both sides by 2:

\[ x = 11 \]

### Conclusion
The required value of \( x \) is \( 11 \).

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