To determine the equation of the line that passes through the points [tex]\((5, 2)\)[/tex], [tex]\((10, 4)\)[/tex], and [tex]\((15, 6)\)[/tex], we need to find the slope (m) and intercept (b) of the line in the form [tex]\(y = mx + b\)[/tex].
Let's follow these steps:
1. Calculate the slope (m) of the line:
The slope between any 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]
Let's take the points [tex]\((5, 2)\)[/tex] and [tex]\((10, 4)\)[/tex]:
[tex]\[
m = \frac{4 - 2}{10 - 5} = \frac{2}{5}
\][/tex]
We can verify it with the points [tex]\((10, 4)\)[/tex] and [tex]\((15, 6)\)[/tex]:
[tex]\[
m = \frac{6 - 4}{15 - 10} = \frac{2}{5}
\][/tex]
Thus, the slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{2}{5}\)[/tex].
2. Find the y-intercept (b):
We can use the point-slope form of the line equation, [tex]\(y = mx + b\)[/tex], and any of the given points to solve for [tex]\(b\)[/tex].
Using the point [tex]\((5, 2)\)[/tex]:
[tex]\[
2 = \frac{2}{5} \cdot 5 + b
\][/tex]
[tex]\[
2 = 2 + b
\][/tex]
Solving for [tex]\(b\)[/tex] gives:
[tex]\[
b = 2 - 2 = 0
\][/tex]
Therefore, the equation of the line is:
[tex]\[
y = \frac{2}{5} x
\][/tex]
3. Compare with the given options:
- Option A: [tex]\(y = \frac{1}{5}x + 1\)[/tex]
- Option B: [tex]\(y = \frac{2}{5}x\)[/tex]
- Option C: [tex]\(y = x - 3\)[/tex]
The correct equation that matches our calculated equation [tex]\(y = \frac{2}{5}x\)[/tex] is:
B. [tex]\(y = \frac{2}{5}x\)[/tex]
