Answer to Question #169344 in Linear Algebra for Mohammad Hossain

Question

Solve the following system of linear equations by the elementary row operations: 5

𝑝 + 2𝑞 − 2𝑟 − 𝑡 = 0

2𝑝 + 5𝑞 − 3𝑟 − 𝑡 = 1

3𝑝 + 8𝑞 − 4𝑟 − 𝑡 = 2

𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3

Answer 1

Solve the following system of linear equations by the elementary row operations:

5𝑝 + 2𝑞 − 2𝑟 − 𝑡 = 0

2𝑝 + 5𝑞 − 3𝑟 − 𝑡 = 1

3𝑝 + 8𝑞 − 4𝑟 − 𝑡 = 2

𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3

Solution:

$𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3\\ 5𝑝 + 2𝑞 − 2𝑟 − 𝑡 = 0\\ 2𝑝 + 5𝑞 − 3𝑟 − 𝑡 = 1\\ 3𝑝 + 8𝑞 − 4𝑟 − 𝑡 = 2$

Multiply 1-st equation by -5 and add it to the second equation

Multiply 1-st equation by -2 and add it to the third equation

Multiply 1-st equation by -3 and add it to the fourth equation

$𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3\\ 0 -23𝑞 − 7𝑟 − 11𝑡 = -15\\ 0 - 5𝑞 − 5𝑟 − 5𝑡 = -5\\ 0 -7𝑞 − 7𝑟 −7 𝑡 = -7$

Multiply 2-nd equation by 23 and add it to the third equation

$𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3\\ 0 +𝑞 +𝑟 +𝑡 = 1\\ 0 -23𝑞 − 7𝑟 − 11𝑡 = -15\\ 0 +0 +0 +0 = 0$

$𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3\\ 0 +𝑞 +𝑟 +𝑡 = 1\\ 0 +0 +16𝑟 +12𝑡 =8\\ 0 +0 +0 +0 = 0$

$𝑝 + 5𝑞 + 𝑟 + 2𝑡 = 3\\ 0 +𝑞 +𝑟 +𝑡 = 1\\ 0 +0 +𝑟 +0.75𝑡 =0.5\\ 0 +0 +0 +0 = 0$

Multiply third equation by -1 and add it to the first equation

Multiply third equation by -1 and add it to the second equation

$𝑝 + 5𝑞 + 0 + 1.25𝑡 = 2.5\\ 0 +𝑞 +0 +0.25𝑡 = 0.5\\ 0 +0 +𝑟 +0.75𝑡 =0.5\\ 0 +0 +0 +0 = 0$

Multiply second equation by -5 and add it to the first equation

$𝑝 +0 + 0 + 0 = 0\\ 0 +𝑞 +0 +0.25𝑡 = 0.5\\ 0 +0 +𝑟 +0.75𝑡 =0.5\\ 0 +0 +0 +0 = 0$

Answer:

$𝑝= 0\\ 𝑞= 0.5-0.25𝑡\\ 𝑟=0.5-0.75𝑡$