how do I solve this math problem? (advanced algebra)?

Question

I know the answer but I want to know how to sovle it.

For what value of n does the system below have infitnetly many solutions?

-3x+2y=7
9x+ny=-21

please help I have an exam tomorrow!!!!

Answer 1

The system has infinitely many solutions if the two lines described by the equations coincide. If you multiply the first with -3 you have 9x-6y=-21. So n must be -6

Answer 2

Two ways you could solve this:

1) Do some substitution to get rid of the "x" term. The simples way to do this is multiply the first equation by 3 and add it to the second equation. This leaves you with (-9x + 9x) + 6y+ny = 21 + (-21), which simplifies to (6+n)y = 0. Notice that if 6+n is zero, then it doesn't matter what value y is because the product is going to be zero. So if n=-6, y can take on an infinite amount of solutions. This means there are an infinite amount of values x can take too, since you can always solve one of the original equations for x and plug any y in.

2) Both are equations of lines, because you can rewrite them in the form of y=mx+b. The two equations will have a single solution if the lines cross in one place, no solution if they never cross (which would only happen if parellel) and an infinite amount of solutions if the two lines perfectly overlap each other. You can rewrite these equations in the form of y=mx+b, and find the value of n that makes the two equations the same.

Answer 3

In order to have infinitely many solutions, one equation must be a linear combination of the other. To put it in more simpler terms, the result of your elimination should yield something which is true.

Your answer should be n = -6.

As for how you solve for it, let's hypothetically use elimination.

Let's multiply the top equation by 3. This will give us these two equations.

-9x + 6y = 21
9x + ny = -21

Now, let's add them together.

6y + ny = 0

What we want is a value for n which will makes this equation ALWAYS true (as this is the basis of having infinitely many solutions). Rearranging the equation a bit

6y = -ny

To make this equation always true, equate the coefficients of y.

6 = -n

Therefore, n = -6.

Answer 4

-3x+2y=7
(+3x)-3x+2y=7+(+3x)
2y=10x
divide both sides by 2 &
get y=5

Solve for x: -3x+2(5x)=7
-3x+10x =7
7x =7
divide both sides by 7 &
get x =1

Now solve for n: 9x+ny=-21
9(1)+n(5)=-21
(-9)+9+5n =-21+(-9)
5n= -30 then divide both sides by 5
n= -6

Answer 5

I think
n=-6

-3x+2y=7
2y=7+3x
2y-7=3x
2/3y-7/3=x

9(2/3y-7/3)+ny=-21
6y-21+ny=-21
6y+ny=0
n=-6

Answer 6

I'm not going to answer your question, I just wanted to say that that is a REALLy smart dog!