To determine which of the points lies on the line described by the equation [tex]\(y - 5 = 6(x - 7)\)[/tex], we need to check each point by substituting the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates into the equation and seeing if it holds true.
Point A: [tex]\((-7, -5)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = -5\)[/tex] into the equation:
[tex]\[
-5 - 5 = 6(-7 - 7)
\][/tex]
Simplify both sides:
[tex]\[
-10 = 6(-14)
\][/tex]
[tex]\[
-10 = -84 \quad \text{(This is false)}
\][/tex]
So, point A does not lie on the line.
Point B: [tex]\((5, 7)\)[/tex]
Substitute [tex]\(x = 5\)[/tex] and [tex]\(y = 7\)[/tex] into the equation:
[tex]\[
7 - 5 = 6(5 - 7)
\][/tex]
Simplify both sides:
[tex]\[
2 = 6(-2)
\][/tex]
[tex]\[
2 = -12 \quad \text{(This is false)}
\][/tex]
So, point B does not lie on the line.
Point C: [tex]\((-5, -7)\)[/tex]
Substitute [tex]\(x = -5\)[/tex] and [tex]\(y = -7\)[/tex] into the equation:
[tex]\[
-7 - 5 = 6(-5 - 7)
\][/tex]
Simplify both sides:
[tex]\[
-12 = 6(-12)
\][/tex]
[tex]\[
-12 = -72 \quad \text{(This is false)}
\][/tex]
So, point C does not lie on the line.
Point D: [tex]\((7, 5)\)[/tex]
Substitute [tex]\(x = 7\)[/tex] and [tex]\(y = 5\)[/tex] into the equation:
[tex]\[
5 - 5 = 6(7 - 7)
\][/tex]
Simplify both sides:
[tex]\[
0 = 6(0)
\][/tex]
[tex]\[
0 = 0 \quad \text{(This is true)}
\][/tex]
So, point D lies on the line.
In conclusion, the point that lies on the line described by the equation [tex]\(y - 5 = 6(x - 7)\)[/tex] is:
D. [tex]\((7, 5)\)[/tex]
