To find the equation of the line passing through the points \((3, 3)\) and \((4, 5)\), follow these steps:
1. Identify the coordinates of the points:
[tex]\[
(x_1, y_1) = (3, 3)
\][/tex]
[tex]\[
(x_2, y_2) = (4, 5)
\][/tex]
2. Calculate the slope (m) of the line:
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the given coordinates:
[tex]\[
m = \frac{5 - 3}{4 - 3} = \frac{2}{1} = 2.0
\][/tex]
3. Use the slope-intercept form \(y = mx + b\) to find the y-intercept (b):
First, substitute one of the points and the slope into the equation. We'll use the point \((3, 3)\):
[tex]\[
3 = 2 * 3 + b
\][/tex]
Simplify and solve for \(b\):
[tex]\[
3 = 6 + b
\][/tex]
[tex]\[
b = 3 - 6
\][/tex]
[tex]\[
b = -3.0
\][/tex]
4. Write the equation of the line:
Now that we have both the slope \(m = 2.0\) and the y-intercept \(b = -3.0\), the equation of the line is:
[tex]\[
y = 2.0x - 3.0
\][/tex]
Therefore, the equation of the line passing through the points \((3, 3)\) and \((4, 5)\) is:
[tex]\[
y = 2.0x - 3.0
\][/tex]
