To determine which of the given points lie on the line defined by the equation [tex]\( x - 3y = 8 \)[/tex], we will substitute the coordinates of each point into the equation and check whether the equation holds true.
1. For the point [tex]\((2, 4)\)[/tex]:
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 4 \)[/tex] into the equation [tex]\( x - 3y = 8 \)[/tex]:
[tex]\[
2 - 3(4) = 2 - 12 = -10
\][/tex]
Since [tex]\(-10 \neq 8\)[/tex], the point [tex]\((2, 4)\)[/tex] does not lie on the line.
2. For the point [tex]\((-1, -3)\)[/tex]:
Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( x - 3y = 8 \)[/tex]:
[tex]\[
-1 - 3(-3) = -1 + 9 = 8
\][/tex]
Since [tex]\( 8 = 8 \)[/tex], the point [tex]\((-1, -3)\)[/tex] lies on the line.
3. For the point [tex]\((-2, 2)\)[/tex]:
Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation [tex]\( x - 3y = 8 \)[/tex]:
[tex]\[
-2 - 3(2) = -2 - 6 = -8
\][/tex]
Since [tex]\(-8 \neq 8\)[/tex], the point [tex]\((-2, 2)\)[/tex] does not lie on the line.
4. For the point [tex]\((-8, 0)\)[/tex]:
Substitute [tex]\( x = -8 \)[/tex] and [tex]\( y = 0 \)[/tex] into the equation [tex]\( x - 3y = 8 \)[/tex]:
[tex]\[
-8 - 3(0) = -8
\][/tex]
Since [tex]\(-8 \neq 8\)[/tex], the point [tex]\((-8, 0)\)[/tex] does not lie on the line.
Therefore, the point that lies on the line [tex]\( x - 3y = 8 \)[/tex] is [tex]\((-1, -3)\)[/tex].
