The only variable with a coefficient of 1 is y in the second equation. Solving that equation for y involves the fewest steps, so we start there.

-4x + y = 17

y = 17 + 4x (add 4x to both sides)

Now, we substitute this for y in the first equation.

3x + 5(17 + 4x) = 19

3x + 85 + 20x = 19 (use the distributive property)

23x = -66 (subtract 85 from both sides, collect terms)

x = -66/23 (divide both sides by 23)

x = -2 20/23 (express as a mixed number)

y = 17 + 4(-66/23) (put x into our equation for y)

y = (23*17 - 4*66)/23 (combine terms)

y = (391 - 264)/23 (simplify)

y = 127/23 (simplify)

y = 5 12/23 (express as a mixed number)

The solution is (x, y) = (-2 20/23, 5 12/23)

Check

3(-2 20/23) + 5(5 12/23) = 19

-8 14/23 + 27 14/23 = 19 (yes)

-4(-2 20/23) + (5 12/23) = 17

11 11/23 + 5 12/23 = 17 (yes)

_____

I ♥ my HP-32SII calculator. It works with fractions directly.

-4x + y = 17

y = 17 + 4x (add 4x to both sides)

Now, we substitute this for y in the first equation.

3x + 5(17 + 4x) = 19

3x + 85 + 20x = 19 (use the distributive property)

23x = -66 (subtract 85 from both sides, collect terms)

x = -66/23 (divide both sides by 23)

x = -2 20/23 (express as a mixed number)

y = 17 + 4(-66/23) (put x into our equation for y)

y = (23*17 - 4*66)/23 (combine terms)

y = (391 - 264)/23 (simplify)

y = 127/23 (simplify)

y = 5 12/23 (express as a mixed number)

The solution is (x, y) = (-2 20/23, 5 12/23)

Check

3(-2 20/23) + 5(5 12/23) = 19

-8 14/23 + 27 14/23 = 19 (yes)

-4(-2 20/23) + (5 12/23) = 17

11 11/23 + 5 12/23 = 17 (yes)

_____

I ♥ my HP-32SII calculator. It works with fractions directly.