Understanding Parallel Lines and Slopes

The core concept here is that parallel lines have the same slope. We need to calculate the slope of the two given lines and set them equal to find the unknown value 'y'.

Calculating Slope

The formula for the slope ($m$) of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Slope of the First Line

The first line passes through the points $(3, y)$ and $(2, 7)$. Let $(x_1, y_1) = (3, y)$ and $(x_2, y_2) = (2, 7)$.

The slope of the first line ($m_1$) is:

$m_1 = \frac{7 - y}{2 - 3}$

$m_1 = \frac{7 - y}{-1}$

Slope of the Second Line

The second line passes through the points $(-1, 4)$ and $(0, 6)$. Let $(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (0, 6)$.

The slope of the second line ($m_2$) is:

$m_2 = \frac{6 - 4}{0 - (-1)}$

$m_2 = \frac{2}{0 + 1}$

$m_2 = \frac{2}{1}$

$m_2 = 2$

Equating Slopes for Parallel Lines

Since the two lines are parallel, their slopes must be equal: $m_1 = m_2$.

So, we have:

$\frac{7 - y}{-1} = 2$

Solving for the Value of y

To find the value of 'y', we solve the equation:

Multiply both sides by -1:

$7 - y = 2 \times (-1)$

$7 - y = -2$

Subtract 7 from both sides:

$-y = -2 - 7$

$-y = -9$

Multiply by -1 to get the value of y:

$y = 9$

Conclusion

Therefore, the value of 'y' that makes the two lines parallel is 9.