To find the equation of the line passing through the points [tex]\((3, 3)\)[/tex] and [tex]\((4, 5)\)[/tex], we'll follow these steps:
1. Determine the slope of the line. The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates [tex]\((x_1, y_1) = (3, 3)\)[/tex] and [tex]\((x_2, y_2) = (4, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 3}{4 - 3} = \frac{2}{1} = 2
\][/tex]
2. Find the y-intercept [tex]\(b\)[/tex]. Using the slope-intercept form of the equation of a line [tex]\(y = mx + b\)[/tex], we can substitute one of the points along with the slope to solve for [tex]\(b\)[/tex]. Using the point [tex]\((3, 3)\)[/tex]:
[tex]\[
3 = 2 \cdot 3 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
3 = 6 + b \\
b = 3 - 6 \\
b = -3
\][/tex]
3. Write the equation of the line. Substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the slope-intercept form:
[tex]\[
y = 2x - 3
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((3, 3)\)[/tex] and [tex]\((4, 5)\)[/tex] is:
[tex]\[ y = 2x - 3 \][/tex]
