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?

1 answer

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

\[ \boxed{-9} \]