What is the solution to the system of equations: 3x + 2y + z = 7, 5x + 5y + 4z = 3, and 3x + 2y + 3z = 1?

To find the solution to the system of equations:

we will use the method of substitution or elimination. Let’s start with the first equation (1) and express z in terms of x and y:

z = 7 - 3x - 2y

Now we will substitute this expression for z into the second equation (2):

5x + 5y + 4(7 - 3x - 2y) = 3

Simplifying this gives:

5x + 5y + 28 - 12x - 8y = 3
-7x - 3y + 28 = 3
-7x - 3y = -25
7x + 3y = 25

This simplifies to:

7x + 3y = 25  (Equation 4)

Next, we substitute z = 7 – 3x – 2y into the third equation (3):

3x + 2y + 3(7 - 3x - 2y) = 1

Simplifying this gives:

3x + 2y + 21 - 9x - 6y = 1
-6x - 4y + 21 = 1
-6x - 4y = -20
3x + 2y = 10  (Equation 5)

Now, we have a new system of equations to solve, which are Equation 4 and Equation 5:

We can use the method of substitution again. From Equation 5, express y in terms of x:

2y = 10 - 3x
y = 5 - 1.5x

Now substitute this expression for y into Equation 4:

7x + 3(5 - 1.5x) = 25

Simplifying gives:

7x + 15 - 4.5x = 25
2.5x + 15 = 25
2.5x = 10
x = 4

Now that we have x, we can substitute it back to find y:

y = 5 - 1.5(4) = 5 - 6 = -1

And finally, substitute x and y back into the expression for z:

z = 7 - 3(4) - 2(-1) = 7 - 12 + 2 = -3

So the solution to the system of equations is:

In conclusion, the solution to the given system of equations is:

(x, y, z) = (4, -1, -3)

You can verify these values by plugging them back into the original equations to ensure they hold true.