To determine which equation's graph has a slope of [tex]\(\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept of [tex]\((0, -2)\)[/tex], we need to convert each of the given equations into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] represents the [tex]\(y\)[/tex]-intercept.
Let's analyze each equation step by step:
1. Equation A: [tex]\(2x - 3y = 6\)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
2x - 3y = 6
\][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
-3y = -2x + 6
\][/tex]
Divide both sides by [tex]\(-3\)[/tex]:
[tex]\[
y = \frac{2}{3}x - 2
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{3}\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-2\)[/tex]. So, this equation matches the given slope and [tex]\(y\)[/tex]-intercept.
2. Equation B: [tex]\(3x - 2y = 6\)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
3x - 2y = 6
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
-2y = -3x + 6
\][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = \frac{3}{2}x - 3
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{2}\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-3\)[/tex]. This does not match the given slope and [tex]\(y\)[/tex]-intercept.
3. Equation C: [tex]\(2x + 3y = -2\)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
2x + 3y = -2
\][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
3y = -2x - 2
\][/tex]
Divide both sides by 3:
[tex]\[
y = -\frac{2}{3}x - \frac{2}{3}
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{2}{3}\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-\frac{2}{3}\)[/tex]. This does not match the given slope and [tex]\(y\)[/tex]-intercept.
4. Equation D: [tex]\(3x + 2y = -2\)[/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[
3x + 2y = -2
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
2y = -3x - 2
\][/tex]
Divide both sides by 2:
[tex]\[
y = -\frac{3}{2}x - 1
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] is [tex]\(-1\)[/tex]. This does not match the given slope and [tex]\(y\)[/tex]-intercept.
Only Equation A, [tex]\(2x - 3y = 6\)[/tex], has the required slope of [tex]\(\frac{2}{3}\)[/tex] and a [tex]\(y\)[/tex]-intercept of [tex]\((0, -2)\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
