Which of the following is an equation of a line that will have points in all the quadrants except the first ?

Options

A. `y=2x`
B. `y=2x+3`
C. `y=2x-3`
D. `y=-2x-3`

Explanation

To determine which equation of a line has points in all quadrants except the first, we need to analyze the given equations. The first quadrant is where both x and y are positive. Therefore, we want to find a line that does not intersect this quadrant.

### Step-by-Step Solution:

1. **Understand the Quadrants**:
- The first quadrant is where both x and y are positive (x > 0, y > 0).
- The second quadrant has x < 0 and y > 0.
- The third quadrant has x < 0 and y < 0.
- The fourth quadrant has x > 0 and y < 0.

2. **Analyze the Given Equations**:
- We need to check each equation to see if it intersects the first quadrant.

3. **Option A: y = x**
- This line has a positive slope and passes through the origin (0,0).
- It will have points in the first quadrant (e.g., (1,1)).
- **Conclusion**: Option A is not valid.

4. **Option B: y = 2x + 3**
- This line has a positive slope (2) and a y-intercept of 3.
- When x = 0, y = 3 (point (0,3)).
- When y = 0, solving for x gives x = -3/2 (point (-3/2, 0)).
- The line will also have points in the first quadrant (e.g., (1,5)).
- **Conclusion**: Option B is not valid.

5. **Option C: y = 2x - 3**
- This line has a positive slope (2) and a y-intercept of -3.
- When x = 0, y = -3 (point (0,-3)).
- When y = 0, solving for x gives x = 3/2 (point (3/2, 0)).
- The line will intersect the first quadrant (e.g., (1,-1)).
- **Conclusion**: Option C is not valid.

6. **Option D: y = -2x - 3**
- This line has a negative slope (-2) and a y-intercept of -3.
- When x = 0, y = -3 (point (0,-3)).
- When y = 0, solving for x gives x = -3/2 (point (-3/2, 0)).
- The line will have points in the second, third, and fourth quadrants but not in the first quadrant.
- **Conclusion**: Option D is valid.

### Final Answer:
The equation of the line that will have points in all quadrants except the first is **Option D: y = -2x - 3**.