To determine which points lie on the line given by the equation [tex]\( y = \frac{2}{3}x - 4 \)[/tex], we'll evaluate each point by substituting the [tex]\( x \)[/tex] value into the equation and checking if the [tex]\( y \)[/tex] value matches.
Let's go through each point one-by-one:
1. Point [tex]\((6,0)\)[/tex]:
Substitute [tex]\( x = 6 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(6) - 4 = 4 - 4 = 0
\][/tex]
Since [tex]\( y = 0 \)[/tex], the point [tex]\((6, 0)\)[/tex] lies on the line.
2. Point [tex]\((-3, -6)\)[/tex]:
Substitute [tex]\( x = -3 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(-3) - 4 = -2 - 4 = -6
\][/tex]
Since [tex]\( y = -6 \)[/tex], the point [tex]\((-3, -6)\)[/tex] lies on the line.
3. Point [tex]\((0, 4)\)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(0) - 4 = 0 - 4 = -4
\][/tex]
Since [tex]\( y \neq 4 \)[/tex], the point [tex]\((0, 4)\)[/tex] does not lie on the line.
4. Point [tex]\((-2, -3)\)[/tex]:
Substitute [tex]\( x = -2 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(-2) - 4 = -\frac{4}{3} - 4 \approx -5.33
\][/tex]
Since [tex]\( y \approx -5.33 \)[/tex], the point [tex]\((-2, -3)\)[/tex] does not lie on the line.
5. Point [tex]\((0, -4)\)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(0) - 4 = 0 - 4 = -4
\][/tex]
Since [tex]\( y = -4 \)[/tex], the point [tex]\((0, -4)\)[/tex] lies on the line.
6. Point [tex]\((4, 0)\)[/tex]:
Substitute [tex]\( x = 4 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(4) - 4 = \frac{8}{3} - 4 = \frac{8}{3} - \frac{12}{3} = -\frac{4}{3}
\][/tex]
Since [tex]\( y \neq 0 \)[/tex], the point [tex]\((4, 0)\)[/tex] does not lie on the line.
7. Point [tex]\((5, 2)\)[/tex]:
Substitute [tex]\( x = 5 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(5) - 4 = \frac{10}{3} - 4 = \frac{10}{3} - \frac{12}{3} = -\frac{2}{3}
\][/tex]
Since [tex]\( y \neq 2 \)[/tex], the point [tex]\((5, 2)\)[/tex] does not lie on the line.
8. Point [tex]\((3, -2)\)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[
y = \frac{2}{3}(3) - 4 = 2 - 4 = -2
\][/tex]
Since [tex]\( y = -2 \)[/tex], the point [tex]\((3, -2)\)[/tex] lies on the line.
So, the points that the line [tex]\( y = \frac{2}{3}x - 4 \)[/tex] goes through are:
- [tex]\((6, 0)\)[/tex]
- [tex]\((-3, -6)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- [tex]\((3, -2)\)[/tex]
