To determine the equation of the line that passes through the points [tex]\((3,6)\)[/tex] and [tex]\((4,10)\)[/tex], we need to follow these steps:
1. Calculate the slope ([tex]\(m\)[/tex]) of the line:
The formula for the slope 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]\((3, 6)\)[/tex] and [tex]\((4, 10)\)[/tex] into the formula:
[tex]\[
m = \frac{10 - 6}{4 - 3} = \frac{4}{1} = 4
\][/tex]
2. Find the y-intercept ([tex]\(b\)[/tex]) using the slope-intercept form of the equation:
The slope-intercept form of the equation of a line is:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\(b\)[/tex], we can use one of the given points. Here, we use the point [tex]\((3, 6)\)[/tex]. Plug in the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(m\)[/tex]:
[tex]\[
6 = 4 \cdot 3 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
6 = 12 + b
\][/tex]
[tex]\[
b = 6 - 12
\][/tex]
[tex]\[
b = -6
\][/tex]
3. Write the equation in function notation [tex]\(f(x)\)[/tex]:
Therefore, substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the equation [tex]\(y = mx + b\)[/tex], we get:
[tex]\[
y = 4x - 6
\][/tex]
In function notation, we write this as:
[tex]\[
f(x) = 4x - 6
\][/tex]
Thus, the equation of the line that passes through the points [tex]\((3, 6)\)[/tex] and [tex]\((4, 10)\)[/tex] is:
[tex]\[
f(x) = 4x - 6
\][/tex]
Among the given options, the correct one is:
[tex]\( y = 4x - 6 \)[/tex].
