Use the Cramer's Method for 2 by 2 systems. Given
ax + by = c
dx + ey = f
which is in our case
4x - 7y = 10
x - y = 7
the solutions are
x = det (c, b, f, e) / det (a, b, d, e),
and
y = det (a, c, d, f) / det (a, b, d, e),
where det (a, b, d, e) = ae - bd,
det (c, b, f, e) = ce - bf,
and
det (a, c, d, f) = af - cd.
In your (first) system the parameters are:
a = 4, b = -7, c = 10, d = 1, e = -1, f = 7, so using the above formulas we will have:
x = [(10)*(-1) - (-7)(7)) = -10+49]=39 / [(4)*(-1) - (-7)*(1)]=3 or x=39/3 or x=13,
and
y = [(4)*(7) - (10)(1)]=18 / [(4*(-1) - (-7)*(1)]=3 or y = 18/3 or y=6.
The other system can be solved in a totally similar manner .
If in this process you ever have det (a, b, d, e)=0, then the system will be either inconsistent (i.e. has no solutions), or will have an infinite number of soultions. In the former case this implies that the two lines representing the system are parallel, and in the latter case it implies the two lines are overlapping.
The Cramer's method generalizes to solving n by n systems of linear equations. Well, I hope this helps!