To determine the correct linear relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the given points [tex]\((2c, 0)\)[/tex], [tex]\((5c, c)\)[/tex], and [tex]\((8c, 2c)\)[/tex], we need to check which of the given equations is satisfied by all three points.
### Given Points:
1. [tex]\( (2c, 0) \)[/tex]
2. [tex]\( (5c, c) \)[/tex]
3. [tex]\( (8c, 2c) \)[/tex]
### Possible Linear Equations:
A) [tex]\(x - 3y = 2c\)[/tex]
B) [tex]\(x - 3y = 8c\)[/tex]
C) [tex]\(3x - y = -8c\)[/tex]
D) [tex]\(3x - y = 11c\)[/tex]
#### Check Equation A: [tex]\(x - 3y = 2c\)[/tex]
1. [tex]\( (2c, 0) \)[/tex]
[tex]\[
2c - 3(0) = 2c \quad \text{is TRUE}
\][/tex]
2. [tex]\( (5c, c) \)[/tex]
[tex]\[
5c - 3(c) = 5c - 3c = 2c \quad \text{is TRUE}
\][/tex]
3. [tex]\( (8c, 2c) \)[/tex]
[tex]\[
8c - 3(2c) = 8c - 6c = 2c \quad \text{is TRUE}
\][/tex]
Since all three points satisfy equation [tex]\(x - 3y = 2c\)[/tex], this equation is a possible candidate.
#### Check Equation B: [tex]\(x - 3y = 8c\)[/tex]
1. [tex]\( (2c, 0) \)[/tex]
[tex]\[
2c - 3(0) = 2c \quad \text{not equal to} \quad 8c \quad \text{is FALSE}
\][/tex]
Since the first point [tex]\((2c, 0)\)[/tex] does not satisfy [tex]\(x - 3y = 8c\)[/tex], this equation is not the correct one.
#### Check Equation C: [tex]\(3x - y = -8c\)[/tex]
1. [tex]\( (2c, 0) \)[/tex]
[tex]\[
3(2c) - 0 = 6c \quad \text{not equal to} \quad -8c \quad \text{is FALSE}
\][/tex]
Since the first point [tex]\((2c, 0)\)[/tex] does not satisfy [tex]\(3x - y = -8c\)[/tex], this equation is not the correct one.
#### Check Equation D: [tex]\(3x - y = 11c\)[/tex]
1. [tex]\((2c, 0)\)[/tex]
[tex]\[
3(2c) - 0 = 6c \quad \text{not equal to} \quad 11c \quad \text{is FALSE}
\][/tex]
Since the first point [tex]\((2c, 0)\)[/tex] does not satisfy [tex]\(3x - y = 11c\)[/tex], this equation is not the correct one.
### Conclusion:
The only equation that is satisfied by all three points [tex]\((2c, 0)\)[/tex], [tex]\((5c, c)\)[/tex], and [tex]\((8c, 2c)\)[/tex] is [tex]\( \boxed{x - 3y = 2c} \)[/tex].
