To solve the problem and find which equation uses the combination of 18 groups of 10 students and 5 groups of 12 students to represent the situation in point-slope form, we need to analyze and rewrite the given options in a standard linear equation format.
The point-slope form of a linear equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
Let's summarize the possible equations derived from the given information and compare them with the standard point-slope form.
### Option 1:
[tex]\[ y - 18 = -\frac{5}{6}(x - 5) \][/tex]
- Here, the point [tex]\((x_1, y_1)\)[/tex] is (5, 18), and the slope [tex]\( m \)[/tex] is [tex]\(-\frac{5}{6}\)[/tex].
### Option 2:
[tex]\[ y - 5 = -\frac{5}{6}(x - 18) \][/tex]
- Here, the point [tex]\((x_1, y_1)\)[/tex] is (18, 5), and the slope [tex]\( m \)[/tex] is [tex]\(-\frac{5}{6}\)[/tex].
### Option 3:
[tex]\[ y + 5 = -\frac{5}{6}(x + 18) \][/tex]
- Here, the point [tex]\((x_1, y_1)\)[/tex] is (-18, -5), and the slope [tex]\( m \)[/tex] is [tex]\(-\frac{5}{6}\)[/tex].
### Option 4:
[tex]\[ y + 18 = -\frac{5}{6}(x + 5) \][/tex]
- Here, the point [tex]\((x_1, y_1)\)[/tex] is (-5, -18), and the slope [tex]\( m \)[/tex] is [tex]\(-\frac{5}{6}\)[/tex].
Given that the problem involves 18 groups of 10 students and 5 groups of 12 students, and considering the interpretations of equations, each option must be evaluated for consistency with the scenario.
The correct equation representing this scenario in point-slope form is:
[tex]\[ \boxed{y - 18 = -\frac{5}{6}(x - 5)} \][/tex]
- This equation uses the point [tex]\((5, 18)\)[/tex] and a slope of [tex]\(-\frac{5}{6}\)[/tex], which aligns correctly with the given groups being represented in point-slope form.
