Solve the following pair of simultaneous equations, $3x + y = 5,$
$2x^2 + xy - y^2 = 5.$

1. Express $y$ in terms of $x$ using the linear equation.

From $3x + y = 5$, we have:
$y = 5 - 3x$

2. Substitute the expression for $y$ into the quadratic equation.

$2x^2 + x(5 - 3x) - (5 - 3x)^2 = 5$

3. Expand and simplify to form a quadratic equation in $x$.

$2x^2 + 5x - 3x^2 - (25 - 30x + 9x^2) = 5$
$-x^2 + 5x - 25 + 30x - 9x^2 = 5$
$-10x^2 + 35x - 25 = 5$
$-10x^2 + 35x - 30 = 0$
Dividing by $-5$:
$2x^2 - 7x + 6 = 0$

4. Solve the quadratic equation for $x$.

$(2x - 3)(x - 2) = 0$
$x = 1.5 \quad \text{or} \quad x = 2$

5. Find the corresponding values of $y$.

When $x = 1.5$:
$y = 5 - 3(1.5) = 5 - 4.5 = 0.5$

When $x = 2$:
$y = 5 - 3(2) = 5 - 6 = -1$