The point whose coordinates are `(4, -2)` lies on a line whose slope is `(3)/(2)`. Which of the following are the coordinates of another point on this line?

Options

A. `(1, 0)`
B. `(2, 1)`
C. `(6, 1)`
D. `(7, 0)`

Answer

To find another point on the line given the point (4, -2) and the slope \( \frac{3}{2} \), we can follow these steps:

### Step 1: Write the point-slope form of the line equation
The point-slope form of a line is given by the equation:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Here, we have:
- \( (x_1, y_1) = (4, -2) \)
- \( m = \frac{3}{2} \)

Substituting these values into the equation gives:
\[ y - (-2) = \frac{3}{2}(x - 4) \]
This simplifies to:
\[ y + 2 = \frac{3}{2}(x - 4) \]

### Step 2: Simplify the equation
Now, we can simplify this equation:
\[ y + 2 = \frac{3}{2}x - \frac{3}{2} \times 4 \]
Calculating \( \frac{3}{2} \times 4 \):
\[ \frac{3}{2} \times 4 = 6 \]
So we have:
\[ y + 2 = \frac{3}{2}x - 6 \]
Subtracting 2 from both sides:
\[ y = \frac{3}{2}x - 8 \]

### Step 3: Rearranging to standard form
To convert this into standard form \( Ax + By + C = 0 \):
\[ \frac{3}{2}x - y - 8 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 3x - 2y - 16 = 0 \]

### Step 4: Testing the given points
Now we need to test the provided points to see which one lies on the line defined by the equation \( 3x - 2y - 16 = 0 \).

1. **Testing point (1, 0)**:
\[ 3(1) - 2(0) - 16 = 3 - 0 - 16 = -13 \quad (\text{not } 0) \]

2. **Testing point (2, 1)**:
\[ 3(2) - 2(1) - 16 = 6 - 2 - 16 = -12 \quad (\text{not } 0) \]

3. **Testing point (6, 1)**:
\[ 3(6) - 2(1) - 16 = 18 - 2 - 16 = 0 \quad (\text{satisfies the equation}) \]

4. **Testing point (7, 0)**:
\[ 3(7) - 2(0) - 16 = 21 - 0 - 16 = 5 \quad (\text{not } 0) \]

### Conclusion
The point (6, 1) satisfies the line equation, so it is another point on the line.

### Final Answer
The coordinates of another point on this line are \( (6, 1) \).

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