Question

3x - 2y + z = 8 and 4x - y + 3z = -1 and 5x + y +2z = -1

So, z = ? and y = ? and x = ?

Please show steps.

Answer 1

To solve the system of linear equations, we can use the elimination method, resulting in the solution x = -9, y = 0.5, and z = 2.6.

Step-by-step explanation:

To solve the system of equations

• 3x - 2y + z = 8,
• 4x - y + 3z = -1,
• 5x + y +2z = -1,

we can use the method of elimination or substitution. This example will demonstrate using the elimination method.

1. Eliminate y from the first two equations by multiplying the second equation by 2 and adding it to the first equation. This gives:
   8x + 5z = 7.
2. Next, eliminate y from the second and third equations by adding them together. This yields:
   9x + 5z = -2.
3. Now we have two equations with two variables:
   8x + 5z = 7,
   9x + 5z = -2. Subtract the first from the second to solve for x:
   x = -9.
4. Substitute x back into one of the two variable equations to find z. For example, in 8x + 5z = 7, substitute x = -9 to get z = 2.6.
5. Finally, substitute x and z back into one of the original equations to find y. Using 3x - 2y + z = 8, we find that y = 0.5.

The solution to the system is x = -9, y = 0.5, and z = 2.6.

Answer 2

To find the values of x, y, and z, we can solve the system of equations given:

Equation 1: 3x - 2y + 2 = 8
Equation 2: 4x - y + 3z = -1
Equation 3: 5x + y + 2z = -1

Let's solve step by step:

Step 1: Solve Equation 1 for x:
3x = 2y - 6
x = (2y - 6)/3

Step 2: Substitute the value of x in Equation 2 and Equation 3:
4x - y + 3z = -1
5x + y + 2z = -1

Substituting x:
4((2y - 6)/3) - y + 3z = -1
5((2y - 6)/3) + y + 2z = -1

Step 3: Simplify the equations:
(8y - 24)/3 - y + 3z = -1
(10y - 30)/3 + y + 2z = -1

Multiplying both sides of each equation by 3 to eliminate the denominators:
8y - 24 - 3y + 9z = -3
10y - 30 + 3y + 6z = -3

Step 4: Simplify further:
5y + 9z = 21
13y + 6z = -27

Step 5: Solve the system of equations formed by the simplified equations:
Multiply the first equation by 6 and the second equation by 9 to eliminate y:
30y + 54z = 126
117y + 54z = -243

Subtract the first equation from the second equation:
117y + 54z - (30y + 54z) = -243 - 126
87y = -369
y = -369/87
y = -4.24

Step 6: Substitute the value of y in either of the simplified equations to find z:
5y + 9z = 21
5(-4.24) + 9z = 21
-21.2 + 9z = 21
9z = 21 + 21.2
9z = 42.2
z = 42.2/9
z = 4.69

Step 7: Substitute the values of y and z into the original Equation 1 to find x:
3x - 2y + 2 = 8
3x - 2(-4.24) + 2 = 8
3x + 8.48 + 2 = 8
3x + 10.48 = 8
3x = 8 - 10.48
3x = -2.48
x = -2.48/3
x = -0.83

Therefore, the values of x, y, and z are approximately:
x ≈ -0.83
y ≈ -4.24
z ≈ 4.69