{16x2 +8xy+y2=36 {3x-y=8

To solve the system of equations given:

1. \( 16x^2 + 8xy + y^2 = 36 \) (Equation 1)
2. \( 3x - y = 8 \) (Equation 2)

We can start by solving Equation 2 for \( y \):

\( y = 3x - 8 \) (Substituting this in Equation 1)

Now, substitute \( y \) in Equation 1:

\[ 16x^2 + 8x(3x - 8) + (3x - 8)^2 = 36 \]

Next, we simplify this. First, calculate \( 8x(3x - 8) \):

\[ 8x(3x - 8) = 24x^2 - 64x \]

Now, calculate \( (3x - 8)^2 \):

\[ (3x - 8)^2 = 9x^2 - 48x + 64 \]

Substituting these back into Equation 1 gives:

\[ 16x^2 + (24x^2 - 64x) + (9x^2 - 48x + 64) = 36 \]

Combine the like terms:

\[ 16x^2 + 24x^2 + 9x^2 - 64x - 48x + 64 = 36 \] \[ 49x^2 - 112x + 64 = 36 \]

Now, subtract 36 from both sides:

\[ 49x^2 - 112x + 28 = 0 \]

Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 49 \), \( b = -112 \), and \( c = 28 \).

Calculate the discriminant:

\[ b^2 - 4ac = (-112)^2 - 4 \cdot 49 \cdot 28 \]

Calculate that step-by-step:

First, find \( (-112)^2 \):

\[ 12444 = 12544 \] Now, find \( 4 \cdot 49 \cdot 28 \):

\[ 4 \cdot 49 = 196 \] \[ 196 \cdot 28 = 5488 \]

Now calculate:

\[ 12544 - 5488 = 7056 \]

Now, since the discriminant is positive, we proceed:

Calculate the square root:

\[ \sqrt{7056} = 84 \]

So, applying the quadratic formula:

\[ x = \frac{112 \pm 84}{98} \]

Now calculate \( x \):

1. \( x = \frac{196}{98} = 2 \)
2. \( x = \frac{28}{98} = \frac{14}{49} = \frac{2}{7} \)

Compute \( y \) for each value of \( x \):

1. For \( x = 2 \):

\[ y = 3(2) - 8 = 6 - 8 = -2 \]

2. For \( x = \frac{2}{7} \):

\[ y = 3 \left(\frac{2}{7}\right) - 8 = \frac{6}{7} - 8 = \frac{6 - 56}{7} = \frac{-50}{7} \]

Thus, the solutions to the system of equations are:

\[ (2, -2) \quad \text{and} \quad \left(\frac{2}{7}, \frac{-50}{7}\right) \]