Answer :
To determine which point lies on the line given by the point-slope equation [tex]\( y - 3 = 4(x + 7) \)[/tex], we will first convert this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex]. This will allow us to easily verify which point satisfies the equation.
### Step-by-Step Solution
1. Starting with the given equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
2. Distribute the 4 on the right-hand side:
[tex]\[ y - 3 = 4x + 28 \][/tex]
3. Add 3 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y - 3 + 3 = 4x + 28 + 3 \][/tex]
Simplifying, we get:
[tex]\[ y = 4x + 31 \][/tex]
Now, we need to determine which of the given points satisfies this equation.
### Checking Each Point
#### Point A: [tex]\( (7, 3) \)[/tex]
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = 3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ 3 = 4(7) + 31 \implies 3 = 28 + 31 \implies 3 = 59 \][/tex]
This is false. Therefore, point [tex]\( (7, 3) \)[/tex] does not satisfy the equation.
#### Point B: [tex]\( (7, -3) \)[/tex]
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ -3 = 4(7) + 31 \implies -3 = 28 + 31 \implies -3 = 59 \][/tex]
This is false. Therefore, point [tex]\( (7, -3) \)[/tex] does not satisfy the equation.
#### Point C: [tex]\( (-7, -3) \)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ -3 = 4(-7) + 31 \implies -3 = -28 + 31 \implies -3 = 3 \][/tex]
This is true. Therefore, point [tex]\( (-7, -3) \)[/tex] satisfies the equation.
#### Point D: [tex]\( (-7, 3) \)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = 3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ 3 = 4(-7) + 31 \implies 3 = -28 + 31 \implies 3 = 3 \][/tex]
This is false. Therefore, point [tex]\( (-7, 3) \)[/tex] does not satisfy the equation.
### Conclusion
The point that lies on the line given by the equation [tex]\( y - 3 = 4(x + 7) \)[/tex] is point [tex]\( C \: (-7, -3) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C. (-7, -3)} \][/tex]
### Step-by-Step Solution
1. Starting with the given equation:
[tex]\[ y - 3 = 4(x + 7) \][/tex]
2. Distribute the 4 on the right-hand side:
[tex]\[ y - 3 = 4x + 28 \][/tex]
3. Add 3 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y - 3 + 3 = 4x + 28 + 3 \][/tex]
Simplifying, we get:
[tex]\[ y = 4x + 31 \][/tex]
Now, we need to determine which of the given points satisfies this equation.
### Checking Each Point
#### Point A: [tex]\( (7, 3) \)[/tex]
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = 3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ 3 = 4(7) + 31 \implies 3 = 28 + 31 \implies 3 = 59 \][/tex]
This is false. Therefore, point [tex]\( (7, 3) \)[/tex] does not satisfy the equation.
#### Point B: [tex]\( (7, -3) \)[/tex]
Substitute [tex]\( x = 7 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ -3 = 4(7) + 31 \implies -3 = 28 + 31 \implies -3 = 59 \][/tex]
This is false. Therefore, point [tex]\( (7, -3) \)[/tex] does not satisfy the equation.
#### Point C: [tex]\( (-7, -3) \)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ -3 = 4(-7) + 31 \implies -3 = -28 + 31 \implies -3 = 3 \][/tex]
This is true. Therefore, point [tex]\( (-7, -3) \)[/tex] satisfies the equation.
#### Point D: [tex]\( (-7, 3) \)[/tex]
Substitute [tex]\( x = -7 \)[/tex] and [tex]\( y = 3 \)[/tex] into the equation [tex]\( y = 4x + 31 \)[/tex]:
[tex]\[ 3 = 4(-7) + 31 \implies 3 = -28 + 31 \implies 3 = 3 \][/tex]
This is false. Therefore, point [tex]\( (-7, 3) \)[/tex] does not satisfy the equation.
### Conclusion
The point that lies on the line given by the equation [tex]\( y - 3 = 4(x + 7) \)[/tex] is point [tex]\( C \: (-7, -3) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C. (-7, -3)} \][/tex]
