valent systems discussionNGSubre3x - 4y - 10Add a Comment:6x + y + 38and-9x + 12y = -30Media Comment9x - 3y = 48SaveThese systems are said to be equivalent Both of the equations in the secondsystem came from the first system somehow.Two questions: How was the first equation in the second system formed fromthe first system? And how was the second equation in the second systemformed from the first system?

We know that these two systems are equivalent, which means that each equation from the second system can be formed as a linear combination of the other two equations from the first system.

From the equation II. a. we have that (with z and w being integer numbers)

z*3x + w*6x = -9x or 3z + 6w = -9

z*(-4y) + w*y = 12y or -4z + w = 12

z*10 + w* 38 = -30 or 10z + 38w = -30

Solving that system for z and w, we discover that z = -3 and w = 0, wich means that equation II. a. is formed by all terms from equation I. a. multiplied by -3.

Doing the same for equation II. b., we have:

z*3x + w*6x = 9x or 3z + 6w = 9

z*(-4y) + w*y = -3y or -4z + w = -3

z*10 + w* 38 = 48 or 10z + 38w = 48

Solving that system for z and w, we discover that z = 1 and w = 1, wich means that equation II. b. is formed by the sum of all terms from the equations I. a. and I. b.