To determine whether the points [tex]\((1, 1.5)\)[/tex] and [tex]\((12, 4)\)[/tex] are solutions to the equation [tex]\(y = \frac{1}{4}x + \frac{5}{4}\)[/tex], we need to substitute the [tex]\(x\)[/tex] values of these points into the equation and check if the resulting [tex]\(y\)[/tex] values match the [tex]\(y\)[/tex] values given in the points.
1. Check the point (1, 1.5):
Substitute [tex]\(x = 1\)[/tex] into the equation:
[tex]\[
y = \frac{1}{4}(1) + \frac{5}{4}
\][/tex]
Simplify the equation:
[tex]\[
y = \frac{1}{4} + \frac{5}{4} = \frac{1 + 5}{4} = \frac{6}{4} = 1.5
\][/tex]
The calculated [tex]\(y\)[/tex] value is 1.5 which matches the given [tex]\(y\)[/tex] value of the point (1, 1.5). Therefore, [tex]\((1, 1.5)\)[/tex] is a solution to the equation.
2. Check the point (12, 4):
Substitute [tex]\(x = 12\)[/tex] into the equation:
[tex]\[
y = \frac{1}{4}(12) + \frac{5}{4}
\][/tex]
Simplify the equation:
[tex]\[
y = \frac{12}{4} + \frac{5}{4} = 3 + \frac{5}{4} = 3 + 1.25 = 4.25
\][/tex]
The calculated [tex]\(y\)[/tex] value is 4.25 which does not match the given [tex]\(y\)[/tex] value of the point (12, 4). Therefore, [tex]\((12, 4)\)[/tex] is not a solution to the equation.
Based on the calculations:
- [tex]\((1, 1.5)\)[/tex] is a solution to the equation.
- [tex]\((12, 4)\)[/tex] is not a solution to the equation.
Therefore, the correct choice is:
[tex]\[
\boxed{(1,1.5) \text{ is a solution to the equation, but } (12, 4) \text{ is not.}}
\][/tex]
