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.