To determine which coordinates lie on the line defined by the equation [tex]\( y = \frac{2}{3} x - 1 \)[/tex], we need to substitute the [tex]\( x \)[/tex] values of each coordinate into the equation and check if the resulting [tex]\( y \)[/tex] values match the [tex]\( y \)[/tex] values of the coordinates.
1. Coordinate (1, 0):
- Substitute [tex]\( x = 1 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3} \cdot 1 - 1 = \frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = \frac{-1}{3}
\][/tex]
- The calculated [tex]\( y \)[/tex] value is [tex]\(-\frac{1}{3}\)[/tex], which does not match the given [tex]\( y = 0 \)[/tex]. Hence, [tex]\((1, 0)\)[/tex] is not on the line.
2. Coordinate (3, 1):
- Substitute [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3} \cdot 3 - 1 = 2 - 1 = 1
\][/tex]
- The calculated [tex]\( y \)[/tex] value is [tex]\(1\)[/tex], which matches the given [tex]\( y = 1 \)[/tex]. Hence, [tex]\((3, 1)\)[/tex] is on the line.
3. Coordinate (6, 3):
- Substitute [tex]\( x = 6 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3} \cdot 6 - 1 = 4 - 1 = 3
\][/tex]
- The calculated [tex]\( y \)[/tex] value is [tex]\(3\)[/tex], which matches the given [tex]\( y = 3 \)[/tex]. Hence, [tex]\((6, 3)\)[/tex] is on the line.
4. Coordinate (-3, -3):
- Substitute [tex]\( x = -3 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3} \cdot (-3) - 1 = -2 - 1 = -3
\][/tex]
- The calculated [tex]\( y \)[/tex] value is [tex]\(-3\)[/tex], which matches the given [tex]\( y = -3 \)[/tex]. Hence, [tex]\((-3, -3)\)[/tex] is on the line.
Therefore, the coordinates that lie on the line [tex]\( y = \frac{2}{3} x - 1 \)[/tex] are [tex]\((3, 1)\)[/tex], [tex]\((6, 3)\)[/tex], and [tex]\((-3, -3)\)[/tex].
