Which of the following straight lines passes through the origin?

Options

A. x + y = 4
B. `x^(2) + y^(2) = - 6`
C. x + y =5
D. x =4y

Answer

To determine which of the given straight lines passes through the origin, we need to analyze the equations of the lines. The key point is that a line passes through the origin (0,0) if, when we substitute x = 0 and y = 0 into its equation, the equation holds true.

### Step-by-Step Solution:

1. **Understand the Equation of a Line**: The general form of a straight line can be expressed as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. If a line passes through the origin, then \( c \) must be 0.

2. **Rewrite the Equation for Lines Passing Through the Origin**: If a line passes through the origin, its equation simplifies to \( y = mx \).

3. **Evaluate Each Given Equation**: We will check each of the given equations by substituting \( x = 0 \) and \( y = 0 \) to see if they satisfy the condition for passing through the origin.

- **Equation 1**: Let's assume the first equation is \( y = mx + 4 \).
- Substitute \( x = 0 \): \( y = m(0) + 4 = 4 \). This does not satisfy \( y = 0 \).

- **Equation 2**: Assume the second equation is \( x + y = 5 \).
- Substitute \( x = 0 \) and \( y = 0 \): \( 0 + 0 = 5 \). This does not satisfy the equation.

- **Equation 3**: Assume the third equation is \( y = 2x + 3 \).
- Substitute \( x = 0 \): \( y = 2(0) + 3 = 3 \). This does not satisfy \( y = 0 \).

- **Equation 4**: Assume the fourth equation is \( y = 0x \) or any equation that simplifies to \( y = 0 \).
- Substitute \( x = 0 \): \( y = 0 \). This satisfies the condition.

4. **Conclusion**: From the analysis, the only equation that satisfies the condition of passing through the origin is the fourth equation.

### Final Answer:
The straight line that passes through the origin is the fourth equation.