To determine which of the given coordinates exists on the line described by the equation [tex]\( y = 3x + 1 \)[/tex], we will test each point by substituting the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values into the equation and checking if the equality holds true.
Let's test each coordinate one by one:
A. [tex]\((2, 6)\)[/tex]
1. Substitute [tex]\( x = 2 \)[/tex] into the equation [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[
y = 3(2) + 1 = 6 + 1 = 7
\][/tex]
2. Compare the computed [tex]\( y \)[/tex] value (which is 7) with the given [tex]\( y \)[/tex] value (which is 6).
[tex]\[
7 \neq 6
\][/tex]
So, [tex]\((2, 6)\)[/tex] does not lie on the line [tex]\( y = 3x + 1 \)[/tex].
B. [tex]\((3, 1)\)[/tex]
1. Substitute [tex]\( x = 3 \)[/tex] into the equation [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[
y = 3(3) + 1 = 9 + 1 = 10
\][/tex]
2. Compare the computed [tex]\( y \)[/tex] value (which is 10) with the given [tex]\( y \)[/tex] value (which is 1).
[tex]\[
10 \neq 1
\][/tex]
So, [tex]\((3, 1)\)[/tex] does not lie on the line [tex]\( y = 3x + 1 \)[/tex].
C. [tex]\((-1, -2)\)[/tex]
1. Substitute [tex]\( x = -1 \)[/tex] into the equation [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[
y = 3(-1) + 1 = -3 + 1 = -2
\][/tex]
2. Compare the computed [tex]\( y \)[/tex] value (which is -2) with the given [tex]\( y \)[/tex] value (which is also -2).
[tex]\[
-2 = -2
\][/tex]
So, [tex]\((-1, -2)\)[/tex] does indeed lie on the line [tex]\( y = 3x + 1 \)[/tex].
D. [tex]\((-3, 4)\)[/tex]
1. Substitute [tex]\( x = -3 \)[/tex] into the equation [tex]\( y = 3x + 1 \)[/tex]:
[tex]\[
y = 3(-3) + 1 = -9 + 1 = -8
\][/tex]
2. Compare the computed [tex]\( y \)[/tex] value (which is -8) with the given [tex]\( y \)[/tex] value (which is 4).
[tex]\[
-8 \neq 4
\][/tex]
So, [tex]\((-3, 4)\)[/tex] does not lie on the line [tex]\( y = 3x + 1 \)[/tex].
Based on the calculations above, the coordinate that lies on the line [tex]\( y = 3x + 1 \)[/tex] is:
C. [tex]\((-1, -2)\)[/tex].
