To solve the given system of equations for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Let's proceed step by step:
### Step 1: Solve the first equation
We have the equation:
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 2y + x = 5 \][/tex]
This simplifies to:
[tex]\[ 2y = 5 - x \][/tex]
[tex]\[ y = \frac{5 - x}{2} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation
We now substitute [tex]\( y = \frac{5 - x}{2} \)[/tex] into the second equation:
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Substituting [tex]\( y \)[/tex]:
[tex]\[ \frac{5 - x}{2} = 2x(x - 3) - 8 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, clear the fraction by multiplying both sides by 2:
[tex]\[ 5 - x = 4x(x - 3) - 16 \][/tex]
Simplify the right side:
[tex]\[ 5 - x = 4x^2 - 12x - 16 \][/tex]
Rearrange all terms to one side to form a quadratic equation:
[tex]\[ 4x^2 - 12x - 16 + x - 5 = 0 \][/tex]
[tex]\[ 4x^2 - 11x - 21 = 0 \][/tex]
### Step 4: Solve the quadratic equation
To solve [tex]\( 4x^2 - 11x - 21 = 0 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 4 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = -21 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4(4)(-21) = 121 + 336 = 457 \][/tex]
Thus:
[tex]\[ x = \frac{11 \pm \sqrt{457}}{8} \][/tex]
### Step 5: Calculate the values
The solutions for [tex]\( x \)[/tex] are:
[tex]\[ x_1 = \frac{11 + \sqrt{457}}{8} \][/tex]
[tex]\[ x_2 = \frac{11 - \sqrt{457}}{8} \][/tex]
### Step 6: Find corresponding [tex]\( y \)[/tex] values
Using [tex]\( y = \frac{5 - x}{2} \)[/tex], we find [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
For [tex]\( x_1 = \frac{11 + \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_1 = \frac{5 - \frac{11 + \sqrt{457}}{8}}{2} = \frac{40 - 11 - \sqrt{457}}{16} = \frac{29 - \sqrt{457}}{16} \][/tex]
For [tex]\( x_2 = \frac{11 - \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_2 = \frac{5 - \frac{11 - \sqrt{457}}{8}}{2} = \frac{40 - 11 + \sqrt{457}}{16} = \frac{29 + \sqrt{457}}{16} \][/tex]
### Final Solution
Therefore, the solutions for [tex]\( (x, y) \)[/tex] are:
[tex]\[ \left( \frac{11 + \sqrt{457}}{8}, \frac{29 - \sqrt{457}}{16} \right) \][/tex]
and
[tex]\[ \left( \frac{11 - \sqrt{457}}{8}, \frac{29 + \sqrt{457}}{16} \right) \][/tex]
