To determine which equation represents the line passing through the points (1, 1) and (5, 6), we 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]
Using the points (1, 1) and (5, 6):
[tex]\[
m = \frac{6 - 1}{5 - 1} = \frac{5}{4}
\][/tex]
2. Apply the slope-point form of a linear equation:
The slope-point form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(m = \frac{5}{4}\)[/tex], [tex]\(x_1 = 1\)[/tex], and [tex]\(y_1 = 1\)[/tex] into the equation:
[tex]\[
y - 1 = \frac{5}{4}(x - 1)
\][/tex]
3. Compare with the given options:
- Option A: [tex]\(y - 1 = \frac{5}{4}(x - 1)\)[/tex]
- Option B: [tex]\(y + 6 = \frac{4}{5}(x + 5)\)[/tex]
- Option C: [tex]\(y - 6 = \frac{4}{5}(x - 5)\)[/tex]
- Option D: [tex]\(y + 1 = \frac{5}{4}(x + 1)\)[/tex]
We see that Option A: [tex]\(y - 1 = \frac{5}{4}(x - 1)\)[/tex] matches the equation derived from the slope-point form.
Therefore, the correct equation that represents a line passing through the points (1, 1) and (5, 6) is:
A. [tex]\(y - 1 = \frac{5}{4}(x - 1)\)[/tex]
