Question

A system of equations is shown below:

x + 3y = 5 (equation 1)
7x – 8y = 6 (equation 2)

A student wants to prove that if equation 2 is kept unchanged and equation 1 is replaced with the sum of equation 1 and a multiple of equation 2, the solution to the new system of equations is the same as the solution to the original system of equations. If equation 2 is multiplied by 1, which of the following steps should the student use for the proof?

Answer 1

Show that the solution to the system of equations 3x + y = 5 and 8x –7y = 6 is the same as the solution to the given system of equations

Show that the solution to the system of equations 8x – 5y = 11 and 7x – 8y = 6 is the same as the solution to the given system of equations

Show that the solution to the system of equations 15x + 13y = 17 and 7x – 8y = 6 is the same as the solution to the given system of equations

Show that the solution to the system of equations –13x + 15y = 17 and 7x – 8y = 6 is the same as the solution to the given system of equations

Answer 2

help?

Answer 3

\[x + 3y = 5 \\ 7x - 8y = 6 \\ a = a'\\ b = b'\] Now assume the top equation is represented by the a,a' and the second by b,b'. We want to show the following system equals the original: \[a + b = a' + b' \\ b = b'\] If we take the top equation substituting b' -> b:
\[a + b' = a' + b'\\b=b'\] Now if we perform the following operation on the top equation \[\ -b' = -b'\] our system equals the original system through a series of legal manipulations \[a=a'\\b=b'\]

Answer 4

what?

Answer 5

i don't understand any of that

Answer 6

All i did was equate the first equation so that the left hand side = a and the right hand side = a'. I did this also for the second equation. The proof requires you to show that two set of linear equations have the same solutions. You can do that by showing that the two sets of equations are actually the same. The question says the following, assume: \[a = a'\\b = b'\] Then it says show that a manipulation of the first equation doesn't actually change the solutions to the system, specifically it says change the first equation to a + 1 * b: \[a + b = a' + b'\\b=b'\] I.e. for the proof to be possible both the top system and this system must be equal. I achieved that by my previous argument. Proofs are difficult because it is not enough to plugin values. If you show that its true for x=1, and y = 3, that doesn't actually mean its true in all cases.

In order to show its true in all cases, we have to stick with variables which could represent any number.

Answer 7

ok, thanks

Answer 8

@mathrulezz help