10x-8y=4, 20x-16y=8

Question

I want this to be solved by elimination! I just wanted to get the hang of this type of Algebra because it's HARD!!!

Answer 1

Hey Holly! To solve this system of equations using elimination, first notice that the original problem contains two equations (10x - 8y = 4 and 20x - 16y = 8) and two unknown variables (x and y). We want to add both equations in a way that will "eliminate" one of the variables.

One way we can do this is by multiplying the first equation (the entire equation, to ensure that our "new" equation is mathematically equivalent to the old one) by -2. This will give us a "new" first equation of -20x + 16y = -8. We can then add these two equations together to "eliminate" a variable:

-20x + 16y = -8

20x - 16y = 8

When we do this, we see that BOTH x and y are "eliminated". Furthermore, we see that if we were to multiply the first equation by -1, it would actually give us the second equation. Therefore, these two equations are actually the SAME line, meaning that this problem is a special case.

Thinking about this algebraically, the two equations are mathematically equivalent and therefore any value of x and y that makes one equation "true" would also make the other equation "true." Thinking about this graphically, the two equations represent the same line on a graph and therefore would intersect at every single point.

Thus, this system of equations has infinite solutions.