The equation of the straight line passing through the point `(3,2)` and perpendicular to the line `y=x` is

Options

A. `x-y=5`
B. `x+y=5`
C. `x+y=1`
D. `x-y=1`

Explanation

To find the equation of the straight line passing through the point (3, 2) and perpendicular to the line \( y = x \), we will follow these steps:

### Step 1: Determine the slope of the given line
The equation of the line \( y = x \) can be rewritten in slope-intercept form as \( y = 1x + 0 \). This shows that the slope (m) of the line \( y = x \) is 1.

**Hint:** The slope of a line in the form \( y = mx + c \) is given by the coefficient \( m \).

### Step 2: Find the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must equal -1. If the slope of the given line is 1, then the slope of the line we are looking for (let's call it \( m_1 \)) can be found using the formula:
\[ m_1 \cdot m = -1 \]
Substituting \( m = 1 \):
\[ m_1 \cdot 1 = -1 \implies m_1 = -1 \]

**Hint:** Remember that perpendicular slopes multiply to -1.

### Step 3: Use the point-slope form of the line equation
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point the line passes through and \( m \) is the slope. Here, \( (x_1, y_1) = (3, 2) \) and \( m = -1 \).

Substituting these values into the point-slope form:
\[ y - 2 = -1(x - 3) \]

**Hint:** The point-slope form is useful for writing the equation of a line when you know a point on the line and the slope.

### Step 4: Simplify the equation
Now, we will simplify the equation:
\[ y - 2 = -1(x - 3) \]
Distributing the -1:
\[ y - 2 = -x + 3 \]
Adding 2 to both sides:
\[ y = -x + 5 \]

**Hint:** Always simplify the equation to get it in the desired form.

### Step 5: Write the equation in standard form
We can rearrange the equation \( y = -x + 5 \) to standard form:
\[ x + y - 5 = 0 \]

**Hint:** Standard form of a line is typically written as \( Ax + By + C = 0 \).

### Final Answer
The equation of the straight line passing through the point (3, 2) and perpendicular to the line \( y = x \) is:
\[ x + y - 5 = 0 \]