To determine whether the points [tex]\((1,2)\)[/tex] and [tex]\((3,2)\)[/tex] are on the same side of the line given by the equation [tex]\(x + 3y + 1 = 0\)[/tex], we need to substitute these points into the equation and analyze the results. Here is the detailed step-by-step solution:
1. Substitute the first point [tex]\((1, 2)\)[/tex] into the equation [tex]\(x + 3y + 1 = 0\)[/tex]:
[tex]\[
\begin{align*}
x + 3y + 1 & \quad\text{where } (x, y) = (1, 2) \\
1 + 3(2) + 1 & = 1 + 6 + 1 \\
& = 8
\end{align*}
\][/tex]
For the point [tex]\((1, 2)\)[/tex], when substituted into the equation, we get a value of [tex]\(8\)[/tex].
2. Substitute the second point [tex]\((3, 2)\)[/tex] into the equation [tex]\(x + 3y + 1 = 0\)[/tex]:
[tex]\[
\begin{align*}
x + 3y + 1 & \quad\text{where } (x, y) = (3, 2) \\
3 + 3(2) + 1 & = 3 + 6 + 1 \\
& = 10
\end{align*}
\][/tex]
For the point [tex]\((3, 2)\)[/tex], when substituted into the equation, we get a value of [tex]\(10\)[/tex].
3. Analyze the signs of the results:
- For the point [tex]\((1, 2)\)[/tex], the result is [tex]\(8\)[/tex], which is positive.
- For the point [tex]\((3, 2)\)[/tex], the result is [tex]\(10\)[/tex], which is also positive.
4. Compare the signs of the results:
Since both results are positive, it means both points [tex]\((1, 2)\)[/tex] and [tex]\((3, 2)\)[/tex] lie on the same side of the line [tex]\(x + 3y + 1 = 0\)[/tex].
Therefore, the points [tex]\((1,2)\)[/tex] and [tex]\((3,2)\)[/tex] are indeed on the same side of the line [tex]\(x + 3y + 1 = 0\)[/tex].
