To determine which point lies on the line with the given point-slope equation [tex]\( y - 2 = 5(x - 6) \)[/tex], we will verify each point by substituting their coordinates into the equation and checking if both sides of the equation are equal.
The given point-slope form is:
[tex]\[ y - 2 = 5(x - 6) \][/tex]
Let's check each point one by one.
### For point A: [tex]\((-6, -2)\)[/tex]
Substitute [tex]\( x = -6 \)[/tex] and [tex]\( y = -2 \)[/tex] into the equation.
[tex]\[ -2 - 2 = 5(-6 - 6) \][/tex]
[tex]\[ -4 = 5(-12) \][/tex]
[tex]\[ -4 = -60 \][/tex]
This is not true, so point A does not lie on the line.
### For point B: [tex]\((-6, 2)\)[/tex]
Substitute [tex]\( x = -6 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation.
[tex]\[ 2 - 2 = 5(-6 - 6) \][/tex]
[tex]\[ 0 = 5(-12) \][/tex]
[tex]\[ 0 = -60 \][/tex]
This is not true, so point B does not lie on the line.
### For point C: [tex]\((6, 2)\)[/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation.
[tex]\[ 2 - 2 = 5(6 - 6) \][/tex]
[tex]\[ 0 = 5(0) \][/tex]
[tex]\[ 0 = 0 \][/tex]
This is true, so point C lies on the line.
### For point D: [tex]\((6, -2)\)[/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -2 \)[/tex] into the equation.
[tex]\[ -2 - 2 = 5(6 - 6) \][/tex]
[tex]\[ -4 = 5(0) \][/tex]
[tex]\[ -4 = 0 \][/tex]
This is not true, so point D does not lie on the line.
Therefore, the point that lies on the line [tex]\( y - 2 = 5(x - 6) \)[/tex] is:
[tex]\[ \boxed{(6, 2)} \][/tex]
