Andrew Rubin
New member
- Joined
- Jun 24, 2019
- Messages
- 22
I am doing a course in matrix methods, and I am currently working on solving system of equations by elimination. Pedagogically, I have attended better courses. Due to lack of explanation of concrete steps, I am stuck and hopefully some of you may help me.
We are to solve a system of equations with a vector [MATH]\textbf{x}[/MATH] of unknowns:
1. [MATH]1x_{1} + 10x_{2} + 9x_{3} = 23[/MATH]2. [MATH]7x_{1} + 7x_{2} + 1x_{3} = 35[/MATH]3. [MATH]7x_{1} + 8x_{2} + 1x_{3} = 37[/MATH]
We are told that we will eliminate the variables one-by-one.
Step 1 is to multiply eq. 1 with -7 and add results to eq. 2 and eq. 3:
1. [MATH](-7)\cdot1x_{1} + (-7)\cdot10\cdot(-7)x_{2} + (-7)\cdot9x_{3} = (-7)\cdot23[/MATH]
1. [MATH]=-7x_{1} + -70x_{2} + -63x_{3} = -161 [/MATH]
We have now eliminated [MATH]x_{1}[/MATH] in eq. 2 and 3:
2. [MATH]-63x_{2} + -63x_{3} = -126[/MATH]3. [MATH]-62x_{2} + -63x_{3} = -124[/MATH]
*Here comes the my two questions*
We are told that step 2 is to "take eq. 2 and add a multiple of it to eq. 3 to eliminate [MATH]x_{2}[/MATH]". Out of nowhere, we are told to add [MATH]-62/63[/MATH] times eq. 2 to eq. 3 to eliminate [MATH]x_{2}[/MATH]. I do not understand why exactly this number is chosen, and my attempt at solving this is not correct:
3. [MATH](-62/63)\cdot(-63)x_{2}+(-62)x_{2} + (-62/63)\cdot(-63)x_{3}+(-63)x_{3} = (-62/63)\cdot(-126)+(-124)[/MATH]3. [MATH] = 0x_{2} + (-1)x_{3} = 0[/MATH]
In the course, this is the result:
After this result, we continue with back-solving the system of equations, starting with [MATH]-62/63x_{3}=0[/MATH]. Since I am unable to understand the second step, I am currently stuck.
To sum up:
Q1: How do we determine that [MATH]-62/63[/MATH] is correct here?
Q2: What am I doing wrong in step 2?
We are to solve a system of equations with a vector [MATH]\textbf{x}[/MATH] of unknowns:
1. [MATH]1x_{1} + 10x_{2} + 9x_{3} = 23[/MATH]2. [MATH]7x_{1} + 7x_{2} + 1x_{3} = 35[/MATH]3. [MATH]7x_{1} + 8x_{2} + 1x_{3} = 37[/MATH]
We are told that we will eliminate the variables one-by-one.
Step 1 is to multiply eq. 1 with -7 and add results to eq. 2 and eq. 3:
1. [MATH](-7)\cdot1x_{1} + (-7)\cdot10\cdot(-7)x_{2} + (-7)\cdot9x_{3} = (-7)\cdot23[/MATH]
1. [MATH]=-7x_{1} + -70x_{2} + -63x_{3} = -161 [/MATH]
We have now eliminated [MATH]x_{1}[/MATH] in eq. 2 and 3:
2. [MATH]-63x_{2} + -63x_{3} = -126[/MATH]3. [MATH]-62x_{2} + -63x_{3} = -124[/MATH]
*Here comes the my two questions*
We are told that step 2 is to "take eq. 2 and add a multiple of it to eq. 3 to eliminate [MATH]x_{2}[/MATH]". Out of nowhere, we are told to add [MATH]-62/63[/MATH] times eq. 2 to eq. 3 to eliminate [MATH]x_{2}[/MATH]. I do not understand why exactly this number is chosen, and my attempt at solving this is not correct:
3. [MATH](-62/63)\cdot(-63)x_{2}+(-62)x_{2} + (-62/63)\cdot(-63)x_{3}+(-63)x_{3} = (-62/63)\cdot(-126)+(-124)[/MATH]3. [MATH] = 0x_{2} + (-1)x_{3} = 0[/MATH]
In the course, this is the result:
After this result, we continue with back-solving the system of equations, starting with [MATH]-62/63x_{3}=0[/MATH]. Since I am unable to understand the second step, I am currently stuck.
To sum up:
Q1: How do we determine that [MATH]-62/63[/MATH] is correct here?
Q2: What am I doing wrong in step 2?
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