Question

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?

Answer 1

So, we have the two following equation systems:

I. a.) 3x - 4y = 10

I. b.) 6x + y = 38

and

II. a.) -9x + 12y = -30

II. b.) 9x - 3y = 48

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.