To determine the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] for the coordinates [tex]\((0, A)\)[/tex] and [tex]\((B, 0)\)[/tex] that lie on the line described by the equation [tex]\(2x - 3y = 6\)[/tex], we can solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex] individually by considering each point separately.
### Finding [tex]\(A\)[/tex]:
1. For the point [tex]\((0, A)\)[/tex], [tex]\(x = 0\)[/tex].
2. Substitute [tex]\(x = 0\)[/tex] into the equation [tex]\(2x - 3y = 6\)[/tex]:
[tex]\[
2(0) - 3y = 6
\][/tex]
3. Simplifying this, we get:
[tex]\[
-3y = 6
\][/tex]
4. Solving for [tex]\(y\)[/tex], we divide both sides by [tex]\(-3\)[/tex]:
[tex]\[
y = -2
\][/tex]
5. Thus, [tex]\(A\)[/tex] is [tex]\(-2\)[/tex].
### Finding [tex]\(B\)[/tex]:
1. For the point [tex]\((B, 0)\)[/tex], [tex]\(y = 0\)[/tex].
2. Substitute [tex]\(y = 0\)[/tex] into the equation [tex]\(2x - 3y = 6\)[/tex]:
[tex]\[
2x - 3(0) = 6
\][/tex]
3. Simplifying this, we get:
[tex]\[
2x = 6
\][/tex]
4. Solving for [tex]\(x\)[/tex], we divide both sides by 2:
[tex]\[
x = 3
\][/tex]
5. Thus, [tex]\(B\)[/tex] is [tex]\(3\)[/tex].
Therefore, the coordinates [tex]\((0, A)\)[/tex] and [tex]\((B, 0)\)[/tex] have the values [tex]\(A = -2\)[/tex] and [tex]\(B = 3\)[/tex] respectively.
