Answer :
To determine which point lies on the line given by the point-slope equation [tex]\(y - 3 = 4(x + 7)\)[/tex], we need to check if each point satisfies the equation.
### Checking Point A: [tex]\((-7, 3)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = 3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ 3 - 3 = 4(-7 + 7) \][/tex]
[tex]\[ 0 = 4 \cdot 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
The equation holds true, so point [tex]\((-7, 3)\)[/tex] lies on the line.
### Checking Point B: [tex]\((7, -3)\)[/tex]
Substitute [tex]\(x = 7\)[/tex] and [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ -3 - 3 = 4(7 + 7) \][/tex]
[tex]\[ -6 = 4 \cdot 14 \][/tex]
[tex]\[ -6 = 56 \][/tex]
The equation does not hold true, so point [tex]\((7, -3)\)[/tex] does not lie on the line.
### Checking Point C: [tex]\((7, 3)\)[/tex]
Substitute [tex]\(x = 7\)[/tex] and [tex]\(y = 3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ 3 - 3 = 4(7 + 7) \][/tex]
[tex]\[ 0 = 4 \cdot 14 \][/tex]
[tex]\[ 0 = 56 \][/tex]
The equation does not hold true, so point [tex]\((7, 3)\)[/tex] does not lie on the line.
### Checking Point D: [tex]\(( -7, -3)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ -3 - 3 = 4(-7 + 7) \][/tex]
[tex]\[ -6 = 4 \cdot 0 \][/tex]
[tex]\[ -6 \neq 0 \][/tex]
The equation does not hold true, so point [tex]\((-7, -3)\)[/tex] does not lie on the line.
### Conclusion
After evaluating all the points, the only point that lies on the line described by the equation [tex]\(y - 3 = 4(x + 7)\)[/tex] is:
[tex]\[ \boxed{(-7, 3)} \][/tex]
### Checking Point A: [tex]\((-7, 3)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = 3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ 3 - 3 = 4(-7 + 7) \][/tex]
[tex]\[ 0 = 4 \cdot 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
The equation holds true, so point [tex]\((-7, 3)\)[/tex] lies on the line.
### Checking Point B: [tex]\((7, -3)\)[/tex]
Substitute [tex]\(x = 7\)[/tex] and [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ -3 - 3 = 4(7 + 7) \][/tex]
[tex]\[ -6 = 4 \cdot 14 \][/tex]
[tex]\[ -6 = 56 \][/tex]
The equation does not hold true, so point [tex]\((7, -3)\)[/tex] does not lie on the line.
### Checking Point C: [tex]\((7, 3)\)[/tex]
Substitute [tex]\(x = 7\)[/tex] and [tex]\(y = 3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ 3 - 3 = 4(7 + 7) \][/tex]
[tex]\[ 0 = 4 \cdot 14 \][/tex]
[tex]\[ 0 = 56 \][/tex]
The equation does not hold true, so point [tex]\((7, 3)\)[/tex] does not lie on the line.
### Checking Point D: [tex]\(( -7, -3)\)[/tex]
Substitute [tex]\(x = -7\)[/tex] and [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
[tex]\[ -3 - 3 = 4(-7 + 7) \][/tex]
[tex]\[ -6 = 4 \cdot 0 \][/tex]
[tex]\[ -6 \neq 0 \][/tex]
The equation does not hold true, so point [tex]\((-7, -3)\)[/tex] does not lie on the line.
### Conclusion
After evaluating all the points, the only point that lies on the line described by the equation [tex]\(y - 3 = 4(x + 7)\)[/tex] is:
[tex]\[ \boxed{(-7, 3)} \][/tex]
