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Solve the following system of equations:

[tex]\[ \begin{cases}
9x + 6y = 12 \\
-18x - 8y = -4
\end{cases} \][/tex]


Sagot :

To solve the given system of linear equations:
[tex]\[ \begin{cases} 9x + 6y = 12 \\ -18x - 8y = -4 \end{cases} \][/tex]

We'll follow a systematic method to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

### Step 1: Simplify the Equations

First, let's simplify the equations if possible. Both equations appear to be in their simplest forms, so we'll proceed as is.

### Step 2: Express One Variable in Terms of Another

To make it easier, we can solve one of the equations for one variable. Let's solve the first equation for [tex]\( y \)[/tex]:

[tex]\[ 9x + 6y = 12 \implies 6y = 12 - 9x \implies y = \frac{12 - 9x}{6} \implies y = 2 - \frac{3}{2}x. \][/tex]

Now we have:
[tex]\[ y = 2 - \frac{3}{2}x. \][/tex]

### Step 3: Substitute Into the Second Equation

Substitute [tex]\( y = 2 - \frac{3}{2}x \)[/tex] into the second equation:

[tex]\[ -18x - 8y = -4 \implies -18x - 8(2 - \frac{3}{2}x) = -4. \][/tex]

Simplify the expression inside the parentheses:

[tex]\[ -18x - 8(2) + 8 \cdot \frac{3}{2}x = -4 \implies -18x - 16 + 12x = -4. \][/tex]

Combine like terms:

[tex]\[ -6x - 16 = -4. \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ -6x = -4 + 16 \implies -6x = 12 \implies x = -2. \][/tex]

So we have found [tex]\( x = -2 \)[/tex].

### Step 4: Solve for the Other Variable

Now, substitute [tex]\( x = -2 \)[/tex] back into the expression for [tex]\( y \)[/tex]:

[tex]\[ y = 2 - \frac{3}{2}(-2) \implies y = 2 + 3 \implies y = 5. \][/tex]

Thus, we have found [tex]\( y = 5 \)[/tex].

### Conclusion

The solution to the system of equations is:

[tex]\[ \boxed{x = -2, \; y = 5} \][/tex]
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