To find the equation of the line that passes through the points [tex]\((5, 2)\)[/tex] and [tex]\((6, 4)\)[/tex], we need to follow these steps:
1. Calculate the slope (m):
The slope [tex]\(m\)[/tex] of a 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]
Given the points [tex]\((5, 2)\)[/tex] and [tex]\((6, 4)\)[/tex]:
[tex]\[
m = \frac{4 - 2}{6 - 5} = \frac{2}{1} = 2
\][/tex]
2. Find the y-intercept (b):
To find the y-intercept, we use the point-slope form of the line equation, [tex]\( y - y_1 = m(x - x_1) \)[/tex], and solve for [tex]\(b\)[/tex], where the line equation is [tex]\( y = mx + b \)[/tex].
Substituting [tex]\(m = 2\)[/tex] and using the point [tex]\((5, 2)\)[/tex]:
[tex]\[
y = mx + b \implies 2 = 2(5) + b
\][/tex]
[tex]\[
2 = 10 + b
\][/tex]
[tex]\[
b = 2 - 10 = -8
\][/tex]
3. Write the equation of the line:
Now that we have the slope [tex]\(m = 2\)[/tex] and the y-intercept [tex]\(b = -8\)[/tex], we can write the equation of the line:
[tex]\[
y = 2x - 8
\][/tex]
Thus, the equation of the line that passes through the points [tex]\((5, 2)\)[/tex] and [tex]\((6, 4)\)[/tex] is:
[tex]\[
\boxed{y = 2x - 8}
\][/tex]
By comparing this with the given options:
a. [tex]\( y = 2x - 8 \)[/tex]
b. [tex]\( y = 4x - 8 \)[/tex]
c. [tex]\( y = 2x + 12 \)[/tex]
d. [tex]\( y = 2x + 2 \)[/tex]
The correct choice is:
[tex]\[
\boxed{a. \ y = 2x - 8}
\][/tex]
