Question

When solving a system of linear equations in two variables, both variables cancel, what does that mean? How do I know if it is a false or true statement?

Answer 1

For example, with this system:

x + y = 12
-x - y = -12
----------------(add)
0 + 0 = 0
0 = 0

The variables added to zero, and the statement left is true. Zero is equal to zero.
This system has an infinite number of solutions because both equations are really the same.

Now look at this example:

x + 3y = 15
x = 14 - 3y

Add 3y to both sides of the bottom equation and write it under the first equation:
x + 3y = 15
x + 3y = 14
------------------(subtract)
0 + 0 = 1
0 = 1

Here the variables added to zero, but you were left with a false statement, 0 = 1. This means there is no solution.

Answer 2

keep in mind that in this set
x + y = 12
-x - y = -12

both equations are equal to each other, the 2nd one is just being multiplied by -1
if both equations are equal then both lines are equal, is just one line pancaked on top of the other

-------------------------------------------------
if we solve this other set
x + 3y = 15
x = 14 - 3y

for "y", we get
y = -1/3 x +5
y =-1/3 x +14/3

notice the slopes are the same, but they differ a bit on the y-intercept
this means that both lines are parallel, same slope and a bit difference
parallel lines do not touch each other thus no solution

Answer 3

TO BE MORE SPECIFIC;
3x - 2y = 6
-6x + 4y = 7

Answer 4

Ok, multiply the first equation by 2 and write the second equation below it.
Then add the two equations together.
Do you get a true statement? If so, there is an infinite number of solutions.
Do you get a false statement? If so, there is no solution.

Answer 5

thank you @mathstudent55 youre a true algerbraic wizard haha

Answer 6

You're welcome, and thanks.