To determine which points lie on the graph of the equation \( y = \frac{3}{2} x - 4 \), we can substitute the \( x \) and \( y \) coordinates of each given point into the equation and see if the resulting equality holds true.
Let's evaluate each given point:
Point A \((-2, -2)\):
1. Substitute \( x = -2 \) into the equation:
[tex]\[
y = \frac{3}{2} (-2) - 4
\][/tex]
2. Calculate the right-hand side:
[tex]\[
y = -3 - 4 = -7
\][/tex]
3. Compare with the \( y \)-coordinate of the point, which is \( -2 \).
[tex]\[
-7 \neq -2
\][/tex]
Thus, point A \((-2, -2)\) does not lie on the graph.
Point B \((-2, -4)\):
1. Substitute \( x = -2 \) into the equation:
[tex]\[
y = \frac{3}{2} (-2) - 4
\][/tex]
2. Calculate the right-hand side:
[tex]\[
y = -3 - 4 = -7
\][/tex]
3. Compare with the \( y \)-coordinate of the point, which is \( -4 \).
[tex]\[
-7 \neq -4
\][/tex]
Thus, point B \((-2, -4)\) does not lie on the graph.
Point C \((2, -1)\):
1. Substitute \( x = 2 \) into the equation:
[tex]\[
y = \frac{3}{2} (2) - 4
\][/tex]
2. Calculate the right-hand side:
[tex]\[
y = 3 - 4 = -1
\][/tex]
3. Compare with the \( y \)-coordinate of the point, which is \( -1 \).
[tex]\[
-1 = -1
\][/tex]
Thus, point C \((2, -1)\) lies on the graph.
Point D \((2, 1)\):
1. Substitute \( x = 2 \) into the equation:
[tex]\[
y = \frac{3}{2} (2) - 4
\][/tex]
2. Calculate the right-hand side:
[tex]\[
y = 3 - 4 = -1
\][/tex]
3. Compare with the \( y \)-coordinate of the point, which is \( 1 \).
[tex]\[
-1 \neq 1
\][/tex]
Thus, point D \((2, 1)\) does not lie on the graph.
So, the point that lies on the graph of the equation \( y = \frac{3}{2} x - 4 \) is:
C) [tex]\((2, -1)\)[/tex].
