Question

For each ordered pair, determine whether it is a solution to the system of equations:
y=-2x-5
6x+3y=-15

(5,0),(2,7),(4,-13),(-3,1)

Answer 1

So we have the equations, let's call them 1 and 2 :
\[y=-2x-5\]
\[6x+3y=-15\]
Now, there are many ways to solve a system of linear equalities, but let's use the most convinient one, but wich is the most convinient one?
Well, substitution method is ideal, because on equation 1, we already have it solved for y, and since both of initial equations must have the same solution, then there's no problem for me to replace the y value of the first to the equation 2, like this:
\[y=-2x-5\]
\[6x+3y=-15\]
\[=> 6x+3(-2x-5)=-15\]
Now, we have converted equation 2, from a equation with two variables, to a single variable equation, with that, we can just solve for "x" and find the value of the variable that satisfies the system, so let's do that:
\[6x+3(-2x-5)=-15\]
\[6x-6x-15=-15\]
\[0=0\]
Oh boy, seems like we stumbled with a rock, but this system of linear equations has no solution, let me show you why:
\[y=-2x-5\]
\[6x+3y=-15\]
If I rewrite them:
\[-2x-y=5\]
\[6x+3y=-15\]
These equations are inverse and proportional, so no matter what method I use, I will never find an c or y alue that satifies the system of equalities.

Answer 2

Unfortunately, the argument above is incorrect...

A solution where the variables vanish and you are left with 0=0 or the equivalent means that the lines are identical and there are infinitely many solutions. Any point on the line is a solution.

A solution where the variables vanish and you are left with 0 = 3 or other such nonsense means that the liens are parallel and there are no solutions.

Answer 3

I see. Thank you!