To write the equation of the line that passes through the points [tex]\((6,2)\)[/tex] and [tex]\((4,1)\)[/tex], follow these steps:
1. Find the slope (m):
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plug in the given points [tex]\((6, 2)\)[/tex] and [tex]\((4, 1)\)[/tex]:
[tex]\[ m = \frac{1 - 2}{4 - 6} = \frac{-1}{-2} = \frac{1}{2} \][/tex]
2. Find the y-intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept, substitute one of the points and the slope into the equation and solve for [tex]\(b\)[/tex].
Using the point [tex]\((6,2)\)[/tex]:
[tex]\[ 2 = \frac{1}{2} \cdot 6 + b \][/tex]
[tex]\[ 2 = 3 + b \][/tex]
[tex]\[ b = 2 - 3 \][/tex]
[tex]\[ b = -1 \][/tex]
3. Write the equation of the line:
Now that we have the slope [tex]\(m = \frac{1}{2}\)[/tex] and the y-intercept [tex]\(b = -1\)[/tex], the equation of the line is:
[tex]\[ y = \frac{1}{2} x - 1 \][/tex]
Thus, the correct equation of the line that passes through the points [tex]\((6, 2)\)[/tex] and [tex]\((4, 1)\)[/tex] is:
[tex]\[ y = \frac{1}{2} x - 1 \][/tex]
So the correct option is:
[tex]\[ y = \frac{1}{2} x - 1 \][/tex]
Which corresponds to the first option in the list provided.
