Solving a system of equations, why aren't the solutions preserved?

$6{x}^{2}+8xy+4{y}^{2}=3$

and

$2{x}^{2}+5xy+3{y}^{2}=2$

Multiply the second by $8$ to get: $16{x}^{2}+40xy+24{y}^{2}=16$

Multiply the first by $5$ to get: $30{x}^{2}+40xy+20{y}^{2}=15$

Subtract the two to get: $14{x}^{2}-4{y}^{2}=-1$

Later, the guy said to disregard his solution because the solutions to the first two equations do not satisfy the third equation. Why does this happen?

$6{x}^{2}+8xy+4{y}^{2}=3$

and

$2{x}^{2}+5xy+3{y}^{2}=2$

Multiply the second by $8$ to get: $16{x}^{2}+40xy+24{y}^{2}=16$

Multiply the first by $5$ to get: $30{x}^{2}+40xy+20{y}^{2}=15$

Subtract the two to get: $14{x}^{2}-4{y}^{2}=-1$

Later, the guy said to disregard his solution because the solutions to the first two equations do not satisfy the third equation. Why does this happen?