Note that subtracting k is the same as adding - k. Similarly, dividing by k is the same as multiplying by 1k.
No, see solution.
Practice makes perfect
The book proves the Elimination Method. In this proof, one of the equations is multiplied and then the equations are added. We are asked if a proof is required when using subtraction and/or division by a constant.
Subtraction
Note that, in the proof provided by the book, the only restriction on the constant k is that it cannot be zero. Thus, it could be negative. This covers subtraction, which is the addition of a negative. Let's consider an example to fully understand this.
2x+2y=4 & (I) 5x+y=6 & (II)
Let's multiply Equation (II) by two, and then subtract it from Equation (I).
We found that x=1. Let's see whether we get the same value for x if we multiply by - 2, and then add the equations. Consider our system one more time.
2x+2y=4 & (I) 5x+y=6 & (II)
As we have already said, this time we will multiply Equation (II) by - 2. Then, we will add the equations.
We can see above that we found the same solution as before. Thus, multiplying by a positive number k and then subtracting the equations is equivalent to multiplying by - k and then adding the equations.
Division
Following a similar reasoning, since k≠0, we can multiply by 1k. This is the same as dividing by k. Multiplication by any number different than zero is proved in the book. Thus, division by any number is also covered.
