To find the equation of the line that passes through the points [tex]\((2, -3)\)[/tex] and [tex]\((-1, 6)\)[/tex], follow these steps:
Step 1: Determine the slope (m) of the line
The slope [tex]\( m \)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given points [tex]\((x_1, y_1) = (2, -3)\)[/tex] and [tex]\((x_2, y_2) = (-1, 6)\)[/tex] into the formula:
[tex]\[
m = \frac{6 - (-3)}{-1 - 2} = \frac{6 + 3}{-1 - 2} = \frac{9}{-3} = -3
\][/tex]
So, the slope [tex]\( m = -3 \)[/tex].
Step 2: Use the point-slope form to find the equation of the line
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute the values [tex]\( m = -3 \)[/tex], [tex]\( x_1 = 2 \)[/tex], and [tex]\( y_1 = -3 \)[/tex] into the point-slope form:
[tex]\[
y - (-3) = -3(x - 2)
\][/tex]
Simplify the equation:
[tex]\[
y + 3 = -3(x - 2)
\][/tex]
[tex]\[
y + 3 = -3x + 6
\][/tex]
[tex]\[
y = -3x + 6 - 3
\][/tex]
[tex]\[
y = -3x + 3
\][/tex]
Step 3: Determine which of the given options matches the derived equation
Compare the derived equation [tex]\( y = -3x + 3 \)[/tex] with the given options:
a) [tex]\( y = -3x + 3 \)[/tex]
b) [tex]\( y = -3x - 3 \)[/tex]
c) [tex]\( y = 3x - 6 \)[/tex]
d) [tex]\( y = 3x - 2 \)[/tex]
The correct option that matches our derived equation is:
[tex]\[
\boxed{a}
\][/tex]
