To solve for the quotient [tex]\(\frac{x}{y}\)[/tex] given the system of equations:
1. [tex]\[-2x + 2 = 9y - 4x\][/tex]
2. [tex]\[2x + y = -6\][/tex]
Let's first work on simplifying and solving these equations.
### Step 1: Simplify the First Equation
Starting with the first equation:
[tex]\[
-2x + 2 = 9y - 4x
\][/tex]
Add [tex]\(4x\)[/tex] to both sides to get:
[tex]\[
2x + 2 = 9y
\][/tex]
Subtract 2 from both sides to isolate terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
2x = 9y - 2
\][/tex]
We can rewrite it for simplicity:
[tex]\[
2x - 9y = -2 \quad \text{(Equation 3)}
\][/tex]
### Step 2: Simplify the Second Equation
Next, consider the second equation:
[tex]\[
2x + y = -6 \quad \text{(Equation 2)}
\][/tex]
### Step 3: Solve the System of Equations
We have the two equations:
1. [tex]\(2x - 9y = -2\)[/tex]
2. [tex]\(2x + y = -6\)[/tex]
Let's solve the system using substitution or elimination. We'll use the substitution method.
#### Isolate [tex]\(y\)[/tex] from Equation 2:
[tex]\[
2x + y = -6
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = -6 - 2x
\][/tex]
#### Substitute [tex]\(y\)[/tex] in Equation 3:
[tex]\[
2x - 9(-6 - 2x) = -2
\][/tex]
Simplify within the brackets:
[tex]\[
2x + 54 + 18x = -2
\][/tex]
Combine like terms:
[tex]\[
20x + 54 = -2
\][/tex]
Subtract 54 from both sides:
[tex]\[
20x = -56
\][/tex]
Divide by 20:
[tex]\[
x = -\frac{56}{20} = -\frac{14}{5}
\][/tex]
#### Substitute [tex]\(x\)[/tex] back into the isolated [tex]\(y\)[/tex] equation:
[tex]\[
y = -6 - 2\left(-\frac{14}{5}\right)
\][/tex]
Simplify the expression:
[tex]\[
y = -6 + \frac{28}{5}
\][/tex]
To combine the terms, convert [tex]\(-6\)[/tex] to a fraction with the same denominator:
[tex]\[
-6 = \frac{-30}{5}
\][/tex]
Thus:
[tex]\[
y = \frac{-30}{5} + \frac{28}{5} = \frac{-30 + 28}{5} = \frac{-2}{5}
\][/tex]
### Step 4: Calculate the Quotient [tex]\(\frac{x}{y}\)[/tex]
Now that we have the values:
[tex]\[
x = -\frac{14}{5}, \quad y = -\frac{2}{5}
\][/tex]
The quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[
\frac{x}{y} = \frac{-\frac{14}{5}}{-\frac{2}{5}} = \frac{14}{5} \times \frac{5}{2} = \frac{14}{2} = 7
\][/tex]
### Conclusion
The value of the quotient [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[
\boxed{7}
\][/tex]
