Answer :
Final answer:
The solution to the system is x₁ = -16/3, x₂ = 15, and x₃ = 125/6.
Explanation:
To solve the system of linear equations:
x₁ − 3x₂ + 4x₃ = −4
3x₁ −7x₂ +7x₃ =−8
−4x₁ +6x₂−x₃= 7
We can use the method of Gaussian elimination. First, we will eliminate x₁ from the second and third equations by multiplying the first equation by 3 and the third equation by -4:
3(x₁ − 3x₂ + 4x₃) = 3(−4)
-4(x₁ − 3x₂ + 4x₃) = -4(7)
This gives us the new system of equations:
3x₁ − 9x₂ + 12x₃ = −12
4x₁ − 12x₂ + 16x₃ = -28
−4x₁ + 6x₂ − x₃ = 7
Now, we can eliminate x₁ from the second and third equations by adding the first equation to each of them:
(3x₁ − 9x₂ + 12x₃) + (4x₁ − 12x₂ + 16x₃) = (−12) + (-28)
(3x₁ − 9x₂ + 12x₃) + (−4x₁ + 6x₂ − x₃) = (−12) + (7)
This simplifies to:
-x₂ + 3x₃ = -40
-3x₂ + 15x₃ = −5
We now have a system of two equations with two variables. We can solve this system using any method, such as substitution or elimination:
-x₂ + 3x₃ = -40 → x₂ = 3x₃ - 40
-3(3x₃ - 40) + 15x₃ = −5
-9x₃ + 120 + 15x₃ = −5
6x₃ = 125
x₃ = 125/6
Substituting this value back into x₂ = 3x₃ - 40:
x₂ = 3(125/6) - 40
x₂ = 90/6
x₂ = 15
Finally, substituting the values of x₂ and x₃ back into x₁ − 3x₂ + 4x₃ = −4:
x₁ − 3(15) + 4(125/6) = −4
x₁ − 45 + 500/6 = −4
x₁ = -16/3
Therefore, the solution to the system of equations is x₁ = -16/3, x₂ = 15, and x₃ = 125/6.
Learn more about Systems of linear equations here:
https://brainly.com/question/17053411
#SPJ11
