there are 12 girls and 18 boys in a class. What is the largest number of groups they can be split up into and have the same number of boys and girls on each team?

This is a system of linear equations. We can solve by elimination. First, we write out the two equations:

3m - 4n = 1

4m - 6n = 5

We want to make the coefficients of one of the variables the same so that we can eliminate it. Let's make the coefficient of m 12 for both equations. To do that for the first, we have to multiply both sides by 4:

12m - 16n = 4

To do that for the second, we have to multiply both sides by 3:

12m - 18n = 15

Put the two equations together:

12m - 16n = 4

12m - 18n = 15

We can subtract the bottom equation from the top and eliminate the 12m to get:

-16n - (-18n) = 4 - 15

2n = -11

n = -5.5

Now that we know n, we can solve for m. We can use one of the original equations and put in the value -5.5 for n and solve for m: 3m - 4 * (-5.5) = 1

3m + 22 = 1

m = -7

and

n = -5.5

3m - 4n = 1​ Equation 1

4m - 6n = 5​ Equation 2

We clear the m variables in the two equations

m = (1+4n)/3 Equation 1

m = (5 +6n)/4 Equation 2

Now we match them

(1+4n)/3 = (5+6n)/4

4(1+4n) = 3(5+6n)

4+16n = 15 +18n

16n - 18n = 15 -4

-2n = 11

n = -11/2

Now we substitute this value in one of the equations

4m -6n = 5 Equation 2

4m -6(-11/2) = 5

4m +66/2 = 5

4m + 33 = 5

m = (5 -33)/4

m = -28/4 = -7

Answer m = -7 and n = -11/2

checking

3*(-7) - 4(-11/2) = 1

-21 +44/2 = 1

-21 +22 = 1 Correct !

4(-7) -6(-11/2) = 5

-28 +66/2 = 5

-28 +33 = 5 Correct!

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