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