To find the equation of the line that passes through the points [tex]\((2, 1)\)[/tex] and [tex]\((5, 4)\)[/tex], we will:
1. Calculate the slope (m) of the line:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_1, y_1) = (2, 1) \)[/tex] and [tex]\( (x_2, y_2) = (5, 4) \)[/tex].
Plugging in the coordinates:
[tex]\[
m = \frac{4 - 1}{5 - 2} = \frac{3}{3} = 1
\][/tex]
2. Use the point-slope form of the equation of a line to find the equation:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\( m = 1 \)[/tex], [tex]\( x_1 = 2 \)[/tex], and [tex]\( y_1 = 1 \)[/tex]:
[tex]\[
y - 1 = 1(x - 2)
\][/tex]
3. Simplify the equation:
Distribute the slope and move the constants to one side:
[tex]\[
y - 1 = x - 2
\][/tex]
Add 1 to both sides:
[tex]\[
y = x - 2 + 1
\][/tex]
Simplify:
[tex]\[
y = x - 1
\][/tex]
So, the equation of the line passing through the points [tex]\((2, 1)\)[/tex] and [tex]\((5, 4)\)[/tex] is:
[tex]\( \boxed{y = x - 1} \)[/tex]
In the given options:
a. [tex]\(y = x + 1\)[/tex] \\
b. [tex]\(y = -x - 1\)[/tex] \\
c. [tex]\(y = x - 1\)[/tex] \\
d. [tex]\(y = x + 3\)[/tex]
The correct answer is [tex]\( \boxed{c} \)[/tex].
