Solve the following system of linear equations by elimination method.3x + 5y + 6z = 7x + 3y – 2z = 52x + 4y + 3z = 8

Sure, let's solve this system of equations step by step using the elimination method.

The system of equations is:

1. 3x + 5y + 6z = 7
2. x + 3y – 2z = 5
3. 2x + 4y + 3z = 8

Step 1: Let's multiply the second equation by 3 and the first equation by 1, then subtract the first equation from the new second equation to eliminate x:

Multiplying the second equation by 3 gives: 3x + 9y - 6z = 15 Subtracting the first equation from this gives: 4y - 12z = 8

So now we have a new system of equations:

1. 3x + 5y + 6z = 7
2. 4y - 12z = 8
3. 2x + 4y + 3z = 8

Step 2: Let's multiply the third equation by 3/2 and the first equation by 1, then subtract the first equation from the new third equation to eliminate x:

Multiplying the third equation by 3/2 gives: 3x + 6y + 4.5z = 12 Subtracting the first equation from this gives: y - 1.5z = 5

So now we have a new system of equations:

1. 3x + 5y + 6z = 7
2. 4y - 12z = 8
3. y - 1.5z = 5

Step 3: Let's multiply the third equation by 4 and the second equation by 1, then add the third equation to the second equation to eliminate y:

Multiplying the third equation by 4 gives: 4y - 6z = 20 Adding this to the second equation gives: -18z = 28

So now we have a new system of equations:

1. 3x + 5y + 6z = 7
2. 4y - 12z = 8
3. -18z = 28

Step 4: Solve the third equation for z:

z = 28 / -18 = -28/18 = -14/9

Step 5: Substitute z = -14/9 into the first and second equations to find x and y:

Substituting into the first equation gives: 3x + 5y - 84/9 = 7, which simplifies to 3x + 5y = 63/9 + 84/9 = 147/9, and further simplifies to 3x + 5y = 49/3 Substituting into the second equation gives: 4y + 168/9 = 8, which simplifies to 4y = 72/9 - 168/9 = -96/9, and further simplifies to y = -24/9 = -8/3

Step 6: Substitute y = -8/3 into the first equation to find x:

3x - 40/3 = 7, which simplifies to 3x = 21 + 40/3 = 63/3 + 40/3 = 103/3, and further simplifies to x = 103/9

So the solution to the system of equations is x = 103/9, y = -8/3, and z = -14/9.