Question

Line MN passes through points M(4, 3) and N(7, 12). If the equation of the line is written in slope-intercept form, y = mx + b, what is the value of b?

Answer 1

To find the equation of the line passing through the points M(4, 3) and N(7, 12), we first need to calculate the slope \(m\) of the line using the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the coordinates of points M and N, where \(M(4, 3)\) corresponds to \((x_1, y_1)\) and \(N(7, 12)\) corresponds to \((x_2, y_2)\):

\[
m = \frac{12 - 3}{7 - 4} = \frac{9}{3} = 3
\]

Now that we have the slope \(m = 3\), we can use the point-slope form of the equation of a line, which is:

\[
y - y_1 = m(x - x_1)
\]

We can use point M(4, 3) for \((x_1, y_1)\):

\[
y - 3 = 3(x - 4)
\]

Distributing the slope on the right side:

\[
y - 3 = 3x - 12
\]

Next, we add 3 to both sides to put the equation in slope-intercept form \(y = mx + b\):

\[
y = 3x - 12 + 3
\]
\[
y = 3x - 9
\]

In the slope-intercept form \(y = mx + b\), we can see that \(b = -9\).

Thus, the value of \(b\) is

\[
\boxed{-9}
\]