To determine which point lies on the line described by the equation \( y + 4 = 4(x - 3) \), we first need to rewrite the equation in the slope-intercept form \( y = mx + b \).
Starting with the given equation:
[tex]\[ y + 4 = 4(x - 3) \][/tex]
First, expand the right-hand side:
[tex]\[ y + 4 = 4x - 12 \][/tex]
Next, isolate \( y \) by subtracting 4 from both sides:
[tex]\[ y = 4x - 12 - 4 \][/tex]
[tex]\[ y = 4x - 16 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = 4x - 16 \][/tex]
Now, we need to check each of the given points to see which one satisfies this equation.
A. \( (3, -4) \)
- Substitute \( x = 3 \) into the equation:
[tex]\[ y = 4(3) - 16 \][/tex]
[tex]\[ y = 12 - 16 \][/tex]
[tex]\[ y = -4 \][/tex]
- The point \( (3, -4) \) satisfies the equation, so it lies on the line.
B. \( (1, 11) \)
- Substitute \( x = 1 \):
[tex]\[ y = 4(1) - 16 \][/tex]
[tex]\[ y = 4 - 16 \][/tex]
[tex]\[ y = -12 \][/tex]
- The point \( (1, 11) \) does not satisfy the equation because 11 is not equal to -12.
C. \( (1, -11) \)
- Substitute \( x = 1 \):
[tex]\[ y = 4(1) - 16 \][/tex]
[tex]\[ y = 4 - 16 \][/tex]
[tex]\[ y = -12 \][/tex]
- The point \( (1, -11) \) does not satisfy the equation because -11 is not equal to -12.
D. \( (0, 0) \)
- Substitute \( x = 0 \):
[tex]\[ y = 4(0) - 16 \][/tex]
[tex]\[ y = 0 - 16 \][/tex]
[tex]\[ y = -16 \][/tex]
- The point \( (0, 0) \) does not satisfy the equation because 0 is not equal to -16.
E. \( (-2, 3) \)
- Substitute \( x = -2 \):
[tex]\[ y = 4(-2) - 16 \][/tex]
[tex]\[ y = -8 - 16 \][/tex]
[tex]\[ y = -24 \][/tex]
- The point \( (-2, 3) \) does not satisfy the equation because 3 is not equal to -24.
F. \( (2, 9) \)
- Substitute \( x = 2 \):
[tex]\[ y = 4(2) - 16 \][/tex]
[tex]\[ y = 8 - 16 \][/tex]
[tex]\[ y = -8 \][/tex]
- The point \( (2, 9) \) does not satisfy the equation because 9 is not equal to -8.
Therefore, the correct answer is:
A. [tex]\( (3, -4) \)[/tex]
