Question

fill in the missing variables in the slope-intercept form, y = mx + b to write the equation of the line passing through the points (1,1) and (3,5).

Another way to write the equation is

Answer 1

To find the equation of the line passing through the points (1, 1) and (3, 5), we will first calculate the slope (m) using the formula:

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

where \((x_1, y_1) = (1, 1)\) and \((x_2, y_2) = (3, 5)\):

\[
m = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2
\]

Now that we have the slope \(m = 2\), we can use one of the points to find the y-intercept (b) using the slope-intercept form of a line, \(y = mx + b\). We will use the point (1, 1):

\[
1 = 2(1) + b
\]

This simplifies to:

\[
1 = 2 + b \\
b = 1 - 2 \\
b = -1
\]

Now that we have both the slope and the y-intercept, we can write the equation of the line:

\[
y = 2x - 1
\]

So, to summarize, the slope-intercept form of the equation for the line passing through the points (1, 1) and (3, 5) is:

\[
y = 2x - 1
\]

Another way to write the equation is in point-slope form, which would be:

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

Using point (1, 1), this gives:

\[
y - 1 = 2(x - 1)
\]

Either form accurately represents the same line.