-5х = y – 5
—2y = x — 21

9514 1404 393

  (x, y) = (-1 2/9, 11 1/9)

Step-by-step explanation:

My favorite solution method is a graphing calculator. It shows the solution to this system of equations is ...

  (x, y) = (-1 2/9, 11 1/9)

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When the solutions are not integers, sometimes solutions involving elimination or substitution get into arithmetic with fractions. Using matrix methods, such as Cramer's Rule can avoid that until the last step. Here, we'll use a "cross multiplication" method.

First we put both equations into general form.

  5x +y -5 = 0

  x +2y -21 = 0

Now, we consider the array of coefficients, with the first repeated as the last:

  5, 1, -5, 5

  1, 2, -21, 1

The differences of cross products of adjacent columns are ...

  d1 = (5)(2) -(1)(1) = 10 -1 = 9

  d2 = (1)(-21) -(2)(-5) = -21 +10 = -11

  d3 = (-5)(1) -(-21)(5) = -5 +105 = 100

The relation ...

  1/d1 = x/d2 = y/d3

specifies how these differences are used to find the solutions.

  x = d2/d1 = -11/9 = -1 2/9

  y = d3/d1 = 100/9 = 11 1/9

The solution is (x, y) = (-1 2/9, 11 1/9).

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