How can we solve the system of equations 4x + 7y + 2z = 0, 3x + 5y + 3z = 9, and 3x + 6y + z = 1?

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Solving the System of Equations

To solve the system of equations given as:

  • 1. 4x + 7y + 2z = 0
  • 2. 3x + 5y + 3z = 9
  • 3. 3x + 6y + z = 1

We can use either the substitution method or the elimination method. Here, we will use the elimination method for clarity.

Step 1: Rewrite the equations

The given equations can be expressed in a standard format:

  • 1. 4x + 7y + 2z = 0 (Equation 1)
  • 2. 3x + 5y + 3z = 9 (Equation 2)
  • 3. 3x + 6y + z = 1 (Equation 3)

Step 2: Eliminate one variable

We’ll start by eliminating z from Equations 2 and 3. We’ll express z in terms of x and y from Equation 3:

z = 1 - 3x - 6y

Now, substitute z into Equation 2:

3x + 5y + 3(1 - 3x - 6y) = 9

Simplifying this gives:

3x + 5y + 3 - 9x - 18y = 9

Combine like terms:

-6x - 13y + 3 = 9

This simplifies to:

-6x - 13y = 6

Thus,

2x + rac{13}{3}y = -1

Step 3: Substitute back

We’ll return to Equation 1 and substitute for z:

4x + 7y + 2(1 - 3x - 6y) = 0

This gives us:

4x + 7y + 2 - 6x - 12y = 0

Combining like terms results in:

-2x - 5y + 2 = 0

Rearranging gives:

2x + 5y = 2

Step 4: Solve for variables

Now we have a system of two equations:

  • 1. 2x + rac{13}{3}y = -1 (Equation A)
  • 2. 2x + 5y = 2 (Equation B)

We can solve them simultaneously. Subtract Equation A from B:

(2x + 5y) - (2x + rac{13}{3}y) = 2 + 1

This leads to:

(5 - rac{13}{3})y = 3

This simplifies to:

-rac{2}{3}y = 3

From which we can solve for y:

y = -rac{9}{2}

Step 5: Substitute back to find x and z

Now substitute y back into Equation B to find x:

2x + 5(-rac{9}{2}) = 2

Solving gives:

2x - rac{45}{2} = 2

Thus:

2x = rac{45}{2} + 2 = rac{49}{2}

Therefore,

x = rac{49}{4}

Finally, substitute the values of x and y back to Equation 3 to solve for z:

3(rac{49}{4}) + 6(-rac{9}{2}) + z = 1

Solving gives us:

z = 1 - rac{147}{4} + 27 = ...

Final solution:

z = ...

At the end, we find that the solution to the system of equations is:

(x, y, z) = 
ight( rac{49}{4}, -rac{9}{2}, ...
ight)

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