A line containing point (2,4) has slope 3. If point P lies on this line , which of the following could be point P ?

Options

A. (3,7)
B. (2,6)
C. (2,7)
D. (3,-1)

Answer

To solve the problem, we need to find the equation of the line that passes through the point (2, 4) with a slope of 3. We will then check which of the given options lies on this line.

### Step 1: Write the equation of the line
The standard form of the equation of a line is given by:

\[ y = mx + c \]

where \( m \) is the slope and \( c \) is the y-intercept. Here, the slope \( m = 3 \).

### Step 2: Substitute the known point to find \( c \)
We know that the line passes through the point (2, 4). We can substitute \( x = 2 \) and \( y = 4 \) into the equation to find \( c \):

\[ 4 = 3(2) + c \]

### Step 3: Solve for \( c \)
Calculating the right side:

\[ 4 = 6 + c \]

Now, isolate \( c \):

\[ c = 4 - 6 \]
\[ c = -2 \]

### Step 4: Write the final equation of the line
Now that we have \( c \), we can write the equation of the line:

\[ y = 3x - 2 \]

### Step 5: Check each option to see if it lies on the line
We will substitute the coordinates of each option into the equation \( y = 3x - 2 \) to see if they satisfy the equation.

#### Option A: (3, 7)
Substituting \( x = 3 \):

\[ y = 3(3) - 2 = 9 - 2 = 7 \]

Since \( y = 7 \), this point lies on the line.

#### Option B: (2, 6)
Substituting \( x = 2 \):

\[ y = 3(2) - 2 = 6 - 2 = 4 \]

Since \( y \neq 6 \), this point does not lie on the line.

#### Option C: (2, 7)
Substituting \( x = 2 \):

\[ y = 3(2) - 2 = 6 - 2 = 4 \]

Since \( y \neq 7 \), this point does not lie on the line.

#### Option D: (3, -1)
Substituting \( x = 3 \):

\[ y = 3(3) - 2 = 9 - 2 = 7 \]

Since \( y \neq -1 \), this point does not lie on the line.

### Conclusion
The only point that lies on the line is option A: (3, 7).