3. Solve the system for x and y. 7x+8y= 26 -12X-7y=-58​

9514 1404 393

Answer:

  (x, y) = (6, -2)

Step-by-step explanation:

My favorite fast and easy way to solve pairs of linear equations is to use a graphing calculator. The one below shows the solution to be ...

  (x, y) = (6, -2)

__

There is no simple way to use substitution or elimination on these equations, so we elect to use the "cross multiplication method." First, we write the equations in "general form", then write the coefficients in two rows, repeating the first coefficient.

  7x +8y -26 = 0

  12x +7y -58 = 0

__

  [tex]\left[\begin{array}{cccc}7&8&-26&7\\12&7&-58&12\end{array}\right][/tex]

The method gets its name from this next step: compute the difference of cross products of pairs of adjacent columns.

  d1 = (7)(7) -(12)(8) = -47

  d2 = (8)(-58) -(7)(-26) = -282

  d3 = (-26)(12) -(-58)(7) = 94

The solution to the equations is given by ...

  1/d1 = x/d2 = y/d3   ⇒  

  x = d2/d1 = -282/-47 = 6

  y = d3/d1 = 94/-47 = -2

The solution is (x, y) = (6, -2).

_____

Additional comment

If you watch videos on this solution method you will notice the 2×4 array of coefficients is formulated slightly differently, and the final equation has the 1/( ) term at the right end of the series of equalities. We judge this variation to be less confusing and easier to remember. YMMV

The work involved in using this method is 6 multiplies, 3 adds, and 2 divides. We believe that is less work than either substitution or elimination when there is no simplification possible.

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