Find the equation of the line which makes equal intercepts on the axis and passes through the point (4, 5).

Options

A. `x+y=9`
B. `x+2y=7`
C. `2x+2y=7`
D. `2x+y=7`

Answer

To find the equation of the line that makes equal intercepts on the axes and passes through the point (4, 5), we can follow these steps:

### Step 1: Understand the concept of equal intercepts
A line that makes equal intercepts on the x-axis and y-axis can be represented in the form:
\[ x + y = a \]
where \( a \) is the length of the intercepts on both axes.

### Step 2: Identify the slope of the line
Since the intercepts are equal, the slope of the line will be -1 (because the line will make a 45-degree angle with both axes). This means that the line will have a negative slope, indicating that as x increases, y decreases.

### Step 3: Use the point-slope form of the line
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Here, the point is (4, 5) and the slope \( m = -1 \).

### Step 4: Substitute the values into the point-slope form
Substituting \( (x_1, y_1) = (4, 5) \) and \( m = -1 \):
\[ y - 5 = -1(x - 4) \]

### Step 5: Simplify the equation
Distributing the -1:
\[ y - 5 = -x + 4 \]
Now, rearranging the equation:
\[ y + x = 4 + 5 \]
\[ x + y = 9 \]

### Step 6: Final equation
Thus, the equation of the line that makes equal intercepts on the axes and passes through the point (4, 5) is:
\[ x + y = 9 \]