Finding values to make a linear system consistent

• Mohamed Abdul
In summary, the conversation focused on finding the conditions for b1, b2, and b3 to make a given matrix consistent using Gaussian elimination. The method used was to deduct multiples of the first row from the second and third rows to make the first element of those rows equal to zero. Additionally, the conversation discussed checking if the system is consistent for specific values of b1, b2, and b3. It was determined that the condition and process used were correct, with the only error being the use of incorrect values in one case.
Mohamed Abdul

Homework Statement

Given the following matrix:

I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when:
a) b1 = 1, b2 = 1, b3 = 3
b) b1 = 1, b2 = 0., b3 = -1
c) b1 = 1, b2 = 2, b3 = 3

Homework Equations

Gaussian elimination method I used here:
http://mathworld.wolfram.com/GaussianElimination.html

The Attempt at a Solution

For the matrix to be consistent, I knew that the number of non-zero rows had to be less than the number of columns. Hence I tried to get the last row to be 0 0 0 | *; however while I managed to get the last row to become 0 0 1, I don't know how to make it the zero row I want.

I'm wondering if my thought process to make this consistent is correct, or if there is another way I can make this system consistent.

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Deduct multiples of the first row from the second and third rows, where the multipliers used are chosen to make the first element of the modified second and third rows be zero.

From there it is easy to make the last row zero (in the first three columns). If it doesn't look easy, post what you get on here.

andrewkirk said:
Deduct multiples of the first row from the second and third rows, where the multipliers used are chosen to make the first element of the modified second and third rows be zero.

From there it is easy to make the last row zero (in the first three columns). If it doesn't look easy, post what you get on here.

This is what I got from simplifying my matrix. But since the rows all have zeroes at different columns, how can I reduce one of the rows to 0 0 0?

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There's an arithmetic error. You have got a sign wrong in the operation you perform on row 3. Fix that and it should all fall into place.

andrewkirk said:
There's an arithmetic error. You have got a sign wrong in the operation you perform on row 3. Fix that and it should all fall into place.

This is the method I used to find out how the system could be consistent. Is my process for determining the condition and testing it for the given values the correct method?

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The condition and the way you are applying it is correct. It looks like you used the wrong values of b in (iii).

andrewkirk said:
The condition and the way you are applying it is correct. It looks like you used the wrong values of b in (iii).
No that was my bad, I copied it down wrong, the actual values were 0 1 and 2.

1. What is a linear system?

A linear system is a set of equations that can be represented graphically as a straight line. Each equation in the system contains two or more variables that are related to each other by a constant coefficient. Solving a linear system involves finding values for the variables that make all the equations in the system true at the same time.

2. Why is it important to find values that make a linear system consistent?

A consistent linear system has a solution, which means that there is a set of values for the variables that satisfies all the equations in the system. This is important because it allows us to solve real-world problems and make predictions based on the relationships between variables represented in the system.

3. How do you determine if a linear system is consistent?

A linear system is consistent if it has one unique solution, meaning that there is a set of values for the variables that satisfies all the equations in the system. This can be determined by graphing the system and seeing if the lines intersect at one point, or by using algebraic methods such as substitution or elimination to solve for the variables.

4. What if a linear system is inconsistent?

If a linear system is inconsistent, it means that there is no set of values for the variables that satisfies all the equations in the system. This can happen when the equations in the system are contradictory or when the lines representing the equations are parallel and do not intersect. In this case, the system has no solution and is said to be inconsistent.

5. Can a linear system have more than one solution?

Yes, a linear system can have infinitely many solutions if the equations in the system are dependent, meaning that one equation can be derived from another. In this case, all the equations represent the same line and there are infinite points of intersection. The system is still consistent, but there is no unique solution as any set of values that satisfies one equation will also satisfy the others.

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