Determine the solution of the following systems of linear equations using elimination. Briefly explain your work.

3x +2y = 19
2x – 3y = -6

triple the first:

9x + 6y = 57
double the 2nd:
4x - 6y = -12
add them:
13x = 45
x = 45/13

sub into first:
3(45/13) + 2y = 19
2y = 19 - 135/13 = 112/13
y = 56/13

I checked the answers, they are correct

Multiply equation 1 by 2 6x+4y=38 multiply equation 2 by 3 6x-9y=18 Subtract them, then 13y=56. So y =56/13 and after substituting, x=45/13

To solve the given system of linear equations using the elimination method, we aim to eliminate one variable by adding or subtracting the equations.

First, we need to make the coefficients of either the x or the y terms the same. In this case, we can eliminate the x term by multiplying the first equation by 2 and the second equation by 3:

Equation 1: 3x + 2y = 19
Equation 2: 2x - 3y = -6

Multiply Equation 1 by 2:
2(3x + 2y) = 2(19)
6x + 4y = 38

Multiply Equation 2 by 3:
3(2x - 3y) = 3(-6)
6x - 9y = -18

Now, we have two equations:

Equation 3: 6x + 4y = 38
Equation 4: 6x - 9y = -18

Next, we'll subtract or add the equations to eliminate one variable. In this case, we'll eliminate the x term by subtracting Equation 4 from Equation 3:

Equation 3 - Equation 4:
(6x + 4y) - (6x - 9y) = 38 - (-18)
6x + 4y - 6x + 9y = 38 + 18
13y = 56

Simplifying further, we get:

13y = 56

To find the value of y, divide both sides of the equation by 13:

y = 56 / 13

Hence, the value of y is y = 4.31 (rounded to two decimal places).

Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use Equation 1:

3x + 2y = 19

Substituting y = 4.31:

3x + 2(4.31) = 19
3x + 8.62 = 19
3x = 19 - 8.62
3x = 10.38

To find the value of x, divide both sides of the equation by 3:

x = 10.38 / 3

Hence, the value of x is x = 3.46 (rounded to two decimal places).

Therefore, the solution to the given system of linear equations is x = 3.46 and y = 4.31.