To determine which of the given points is a solution to the linear equation [tex]\(2x + 5y - 1 = 0\)[/tex], we need to check each point by substituting the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values into the equation and see if the resulting equation holds true.
Let's analyze each point step-by-step:
1. First point [tex]\((-1, 3)\)[/tex]:
[tex]\[
2(-1) + 5(3) - 1 = 2 \cdot (-1) + 5 \cdot 3 - 1
\][/tex]
Simplify the left-hand side:
[tex]\[
= -2 + 15 - 1
= 12
\][/tex]
Since [tex]\(12 \neq 0\)[/tex], [tex]\((-1, 3)\)[/tex] is not a solution to the equation.
2. Second point [tex]\((2, 1)\)[/tex]:
[tex]\[
2(2) + 5(1) - 1 = 2 \cdot 2 + 5 \cdot 1 - 1
\][/tex]
Simplify the left-hand side:
[tex]\[
= 4 + 5 - 1
= 8
\][/tex]
Since [tex]\(8 \neq 0\)[/tex], [tex]\((2, 1)\)[/tex] is not a solution to the equation.
3. Third point [tex]\((3, -1)\)[/tex]:
[tex]\[
2(3) + 5(-1) - 1 = 2 \cdot 3 + 5 \cdot (-1) - 1
\][/tex]
Simplify the left-hand side:
[tex]\[
= 6 - 5 - 1
= 0
\][/tex]
Since [tex]\(0 = 0\)[/tex], [tex]\((3, -1)\)[/tex] is a solution to the equation.
4. Fourth point [tex]\((-2, 4)\)[/tex]:
[tex]\[
2(-2) + 5(4) - 1 = 2 \cdot (-2) + 5 \cdot 4 - 1
\][/tex]
Simplify the left-hand side:
[tex]\[
= -4 + 20 - 1
= 15
\][/tex]
Since [tex]\(15 \neq 0\)[/tex], [tex]\((-2, 4)\)[/tex] is not a solution to the equation.
Based on the above steps, the point [tex]\((3, -1)\)[/tex] satisfies the equation [tex]\(2x + 5y - 1 = 0\)[/tex]. Therefore, the correct answer is:
(iii) [tex]\((3, -1)\)[/tex]
