Substitution and elimination

Question

Determine the value of n that makes a system of equations with a solution that has a y value of 2.

5x+6y=32

2x+ny=18

Answer 1

Let's define our two equations as eq. #1 and eq. #2:

5x + 6y = 32 (eq. #1)

2x + ny = 18 (eq. #2)

We are trying to determine the value of "n", such that y=2 is a solution to the system of equations. We can use linear combinations and add the two equations in such a way that "x" is eliminated. Multiply eq. #1 by a factor of -2 and multiply eq. # by a factor of 5 (make sure you multiply both sides of the equations).

-10x - 12y = -64 (eq. #1 multiplied by -2)

10x + 5ny = 90 (eq. #2 multiplied by 5)

Add the two above lines to get the following:

5ny - 12y = 26, (factor out a "y" from the left side)

y(5n - 12) = 26

At this point, we have a few options that will get us to the same answer. Option 1 is we know y=2 is a solution to our system of equation, so let's plug that into our expression now.

2(5n - 12) = 26

5n - 12 = 13

5n = 25

n = 5

Option 2 is we could solve this all the way down to "n = ...." and then substitute y=2 at the very end.

5n - 12 = 26/y

5n = (26/y) + 12

n = [(26/y) + 12]/5, (now plug in y = 2)

n = [(26/2) + 12]/5

n = (13 + 12)/5

n = 25/5

n = 5

Also, I believe we could have plugged in y=2 at the very beginning, since we know it is a solution. We could have also done the following

5x + 6(2) = 32

5x + 12 = 32

5x = 20

x = 4, (plug this into eq. #2)

2(4) + n(2) = 18

8 + 2n = 18

2n = 10

n = 5

Lots of options for this one!

Answer 2

We want to find a value of n that results with y = 2

Plug 2 in for y into the first equation (because it doesn't have n)

5x + 6(2) = 32

5x + 12 = 32

5x = 20

x = 4

Now substitute x and y in the second equation and solve for n.

2x + ny = 18

2(4) + n(2) = 18

8 + 2n = 18

2n = 10

n = 5

5x+6y=32

2x+5y=18

Answer 3

This actually not to bad because you are given the value of y.

1st step: substitute 2 for y in both equations then solve:

5x + 6y = 32 2x + ny =18

5x + 6(2) = 32 2x + 2n = 18

5x + 12 = 32

5x = 20

x = 4

2nd step: Now you have the value for y and x. Time to plug both into the second equation to solve for n.

2x + ny = 18

2(4) + 2n = 18

8 + 2n = 18

2n = 10

n = 5

I hope this helped you.