Which of the following could be the equation of a line that passes through the points (-2,-7) and (2,17) in the standard (x,y) coordinate plane ?

Options

A. 3x-2y=8
B. 6x-y=-5
C. 5x-2y=7
D. 9x-2y=-16

Answer

To find the equation of the line that passes through the points (-2, -7) and (2, 17), we can use the two-point form of the equation of a line. The steps are as follows:

### Step 1: Identify the points
We have two points:
- Point 1: \( (x_1, y_1) = (-2, -7) \)
- Point 2: \( (x_2, y_2) = (2, 17) \)

### Step 2: Use the two-point form of the line equation
The two-point form of the equation of a line is given by:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]

### Step 3: Substitute the values into the equation
Substituting the values of the points into the equation:
\[ \frac{y - (-7)}{17 - (-7)} = \frac{x - (-2)}{2 - (-2)} \]
This simplifies to:
\[ \frac{y + 7}{17 + 7} = \frac{x + 2}{2 + 2} \]
\[ \frac{y + 7}{24} = \frac{x + 2}{4} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 4(y + 7) = 24(x + 2) \]

### Step 5: Distribute both sides
Distributing both sides results in:
\[ 4y + 28 = 24x + 48 \]

### Step 6: Rearrange the equation
Rearranging the equation to standard form (Ax + By = C):
\[ 4y = 24x + 48 - 28 \]
\[ 4y = 24x + 20 \]
Now, dividing the entire equation by 4:
\[ y = 6x + 5 \]

### Step 7: Convert to standard form
To convert this to standard form, we rearrange:
\[ 6x - y = -5 \]
This can also be written as:
\[ 6x - y = -5 \]

### Step 8: Identify the correct option
Now, comparing this equation with the options provided:
- Option 1: \( 3x - 2y = 8 \)
- Option 2: \( 6x - y = -5 \) (This is our answer)
- Option 3: \( 5x - 2y = 7 \)
- Option 4: \( 9x - 2y = -16 \)

The correct answer is **Option 2: \( 6x - y = -5 \)**.

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