A student obtained the two regression lines ` 6x - 15y -21 =0 and 21x + 14y - 56 =0 ` Is he found the equations correct ? Justify.

Answer

To determine if the two regression lines given by the equations \(6x - 15y - 21 = 0\) and \(21x + 14y - 56 = 0\) are correct, we need to analyze the regression coefficients derived from these equations.

### Step 1: Rearranging the first regression line
The first regression line is given by:
\[ 6x - 15y - 21 = 0 \]
We can rearrange this equation to express \(y\) in terms of \(x\):
\[ 6x = 15y + 21 \]
\[ 15y = 6x - 21 \]
\[ y = \frac{6}{15}x - \frac{21}{15} \]
This simplifies to:
\[ y = \frac{2}{5}x - \frac{7}{5} \]
From this, we can identify the regression coefficient \(b_{xy}\):
\[ b_{xy} = \frac{2}{5} \]

### Step 2: Rearranging the second regression line
The second regression line is given by:
\[ 21x + 14y - 56 = 0 \]
We can rearrange this equation to express \(y\) in terms of \(x\):
\[ 14y = -21x + 56 \]
\[ y = -\frac{21}{14}x + \frac{56}{14} \]
This simplifies to:
\[ y = -\frac{3}{2}x + 4 \]
From this, we can identify the regression coefficient \(b_{yx}\):
\[ b_{yx} = -\frac{3}{2} \]

### Step 3: Analyzing the regression coefficients
Now we have:
- \(b_{xy} = \frac{2}{5}\)
- \(b_{yx} = -\frac{3}{2}\)

### Step 4: Checking the signs of the regression coefficients
In regression analysis, the signs of the regression coefficients should be the same. Here, \(b_{xy}\) is positive and \(b_{yx}\) is negative. Therefore, the signs are different.

### Conclusion
Since the signs of the regression coefficients \(b_{xy}\) and \(b_{yx}\) are different, the equations obtained by the student are incorrect.

### Final Answer
The student did not find the equations correct because the signs of the regression coefficients are different.

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