To find the equation of the line that passes through the points [tex]\((1, -3)\)[/tex] and [tex]\((4, 6)\)[/tex], we need to determine the slope of the line and its y-intercept.
### Step 1: Calculating the Slope
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:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((1, -3)\)[/tex] and [tex]\((4, 6)\)[/tex], we get:
[tex]\[
m = \frac{6 - (-3)}{4 - 1} = \frac{6 + 3}{4 - 1} = \frac{9}{3} = 3
\][/tex]
### Step 2: Finding the Y-Intercept
The equation of a line in slope-intercept form is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We can use one of the given points to find [tex]\( b \)[/tex]. Let's use the point [tex]\((1, -3)\)[/tex]:
[tex]\[
-3 = 3(1) + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
-3 = 3 + b \implies b = -3 - 3 \implies b = -6
\][/tex]
### Step 3: Writing the Equation of the Line
Given the slope [tex]\( m = 3 \)[/tex] and the y-intercept [tex]\( b = -6 \)[/tex], the equation of the line is:
[tex]\[
y = 3x - 6
\][/tex]
### Step 4: Matching with the Given Options
The equation we derived is [tex]\( y = 3x - 6 \)[/tex]. Checking the provided options:
a) [tex]\( y = -3x - 3 \)[/tex]
b) [tex]\( y = 3x - 6 \)[/tex]
c) [tex]\( y = -3x + 3 \)[/tex]
d) [tex]\( y = 3x - 2 \)[/tex]
The correct answer is option b) [tex]\( y = 3x - 6 \)[/tex].
