Start by putting both equations in standard form.

[first equation]

2(x-y) = 3+x

2x - 2y - x = 3 (expand the parentheses, subtract x from both sides)

x - 2y = 3 (collect terms)

[second equation]

x = 3y + 4

x - 3y = 4 (subtract 3y from both sides)

Now we can see that the coefficient of x is the same for both equations, so we can do "elimination" by subtracting one equation from the other. We elect to subtract the second equation from the first, so the coefficient of y will end up positive.

(x - 2y) - (x - 3y) = (3) - (4)

x - x - 2y + 3y = -1 (rearrange terms on the left, evaluate on the right)

y = -1 (collect terms. Solution obtained by elimination.)

We can use the second original equation to find x given this value of y

x = 3(-1) + 4

x = -3 + 4

x = 1 (Solution obtained by substitution.)

If we want to use "elimination" to find x, we can subtract 2 times the second (standard form) equation from 3 times the first equation. These multipliers come from the coefficients of y in the two equations. The idea is to choose a combination of the two equations that will eliminate y.

3(x - 2y) - 2(x - 3y) = 3(3) - 2(4)

3x - 6y - 2x + 6y = 9 - 8 (multiply everything out)

x = 1 (collect terms. Solution obtained by elimination.)

The solution using elimination is (x, y) = (1, -1).

Check

2((1)-(-1)) = 3 + (1), (1) = 3(-1) + 4

2(2) = 4, 1 = -3 + 4 (yes for both)

[first equation]

2(x-y) = 3+x

2x - 2y - x = 3 (expand the parentheses, subtract x from both sides)

x - 2y = 3 (collect terms)

[second equation]

x = 3y + 4

x - 3y = 4 (subtract 3y from both sides)

Now we can see that the coefficient of x is the same for both equations, so we can do "elimination" by subtracting one equation from the other. We elect to subtract the second equation from the first, so the coefficient of y will end up positive.

(x - 2y) - (x - 3y) = (3) - (4)

x - x - 2y + 3y = -1 (rearrange terms on the left, evaluate on the right)

y = -1 (collect terms. Solution obtained by elimination.)

We can use the second original equation to find x given this value of y

x = 3(-1) + 4

x = -3 + 4

x = 1 (Solution obtained by substitution.)

If we want to use "elimination" to find x, we can subtract 2 times the second (standard form) equation from 3 times the first equation. These multipliers come from the coefficients of y in the two equations. The idea is to choose a combination of the two equations that will eliminate y.

3(x - 2y) - 2(x - 3y) = 3(3) - 2(4)

3x - 6y - 2x + 6y = 9 - 8 (multiply everything out)

x = 1 (collect terms. Solution obtained by elimination.)

The solution using elimination is (x, y) = (1, -1).

Check

2((1)-(-1)) = 3 + (1), (1) = 3(-1) + 4

2(2) = 4, 1 = -3 + 4 (yes for both)