Let X and Y be two related variables. The two regression lines are given by x-y+ 1=0 and 2x-y +4=0. The two regression lines pass through the point

Options

A. (-4,-3)
B. (-6,-5)
C. (3,-2)
D. (-3,-2)

Answer

To determine the point through which the two regression lines pass, we need to solve the equations of the regression lines given:

1. **Equation 1**: \( x - y + 1 = 0 \)
2. **Equation 2**: \( 2x - y + 4 = 0 \)

### Step 1: Rewrite the equations in slope-intercept form

For the first equation:
\[ x - y + 1 = 0 \implies y = x + 1 \]

For the second equation:
\[ 2x - y + 4 = 0 \implies y = 2x + 4 \]

### Step 2: Set the equations equal to find the intersection point

Now, we set the two equations equal to each other to find the point of intersection:
\[ x + 1 = 2x + 4 \]

### Step 3: Solve for \(x\)

Rearranging the equation:
\[ x + 1 - 2x - 4 = 0 \implies -x - 3 = 0 \implies x = -3 \]

### Step 4: Substitute \(x\) back to find \(y\)

Now, substitute \(x = -3\) back into one of the equations to find \(y\). We can use the first equation:
\[ y = -3 + 1 = -2 \]

### Conclusion

The point through which both regression lines pass is \((-3, -2)\).

### Final Answer

The two regression lines pass through the point \((-3, -2)\), which corresponds to option 4.

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