In the xy-cordinate system if (a,b) and `(a + 3, b + k)` are two points on the line defined by the equaiton `x = 3y- 7,` then k =

Options

A. 9
B. 3
C. `7/3`
D. 1

Answer

To solve the problem, we need to find the value of \( k \) given that the points \( (a, b) \) and \( (a + 3, b + k) \) lie on the line defined by the equation \( x = 3y - 7 \).

### Step-by-step Solution:

1. **Substitute the first point \( (a, b) \) into the line equation:**
\[ x = a \quad \text{and} \quad y = b \]
Substituting these into the equation \( x = 3y - 7 \):
\[ a = 3b - 7 \]
This gives us our first equation.

**Hint:** Use the line equation to express \( a \) in terms of \( b \).

2. **Substitute the second point \( (a + 3, b + k) \) into the line equation:**
\[ x = a + 3 \quad \text{and} \quad y = b + k \]
Substituting these into the line equation \( x = 3y - 7 \):
\[ a + 3 = 3(b + k) - 7 \]
Simplifying the right side:
\[ a + 3 = 3b + 3k - 7 \]

**Hint:** Make sure to distribute the 3 correctly when substituting for \( y \).

3. **Set up the equation:**
Now we have two equations:
- From step 1: \( a = 3b - 7 \)
- From step 2: \( a + 3 = 3b + 3k - 7 \)

Substitute \( a \) from the first equation into the second equation:
\[ (3b - 7) + 3 = 3b + 3k - 7 \]

**Hint:** Substitute carefully to eliminate \( a \) from the second equation.

4. **Simplify the equation:**
Simplifying the left side:
\[ 3b - 4 = 3b + 3k - 7 \]
Now, cancel \( 3b \) from both sides:
\[ -4 = 3k - 7 \]

**Hint:** Look for terms that can be canceled out to simplify the equation.

5. **Solve for \( k \):**
Rearranging gives:
\[ 3k = -4 + 7 \]
\[ 3k = 3 \]
Dividing both sides by 3:
\[ k = 1 \]

**Hint:** Isolate \( k \) by performing inverse operations.

### Final Answer:
The value of \( k \) is \( 1 \).