Solve Systems of Equations Using Matrices and Row Operations

Math Problem Statement

Homework:12.2 HW - Systems of Linear Equations: Matrices

Question 23, 12.2.55

HW Score: 57.86%, 20.25 of 35 points

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Part 1

Solve the following system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

left brace Start 3 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column 3rd Row 1st Column EndMatrix negative x−x

plus+

equals=

negative 4−4

negative x−x

plus+

5 y5y

minus−

23 z23z

equals=

negative 32−32

7 x7x

minus−

3 y3y

minus−

31 z31z

equals=

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Part 1

Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

The solution is

left parenthesis nothing comma nothing comma nothing right parenthesisenter your response here,enter your response here,enter your response here.

(Simplify your answers.)

There are infinitely many solutions. The solution can be written as

StartSet left parenthesis x comma y comma z right parenthesis font size decreased by 6 | font size decreased by 6 x equals nothing comma y equals nothing comma z is any real number EndSet(x,y,z) | x=enter your response here, y=enter your response here, z is any real number.

(Simplify your answers. Type expressions using z as the variable.)

There are infinitely many solutions. The solution can be written as

StartSet left parenthesis x comma y comma z right parenthesis font size decreased by 6 | font size decreased by 6 x equals nothing comma y is any real number comma z is any real number EndSet(x,y,z) | x=enter your response here, y is any real number, z is any real number.

(Simplify your answer. Type an expression using y and z as the variables.)

The system is inconsistent.

If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

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Solution

To solve this system of equations using matrices and row operations, we need to represent it in augmented matrix form and perform row operations to achieve row echelon form or reduced row echelon form.

The system given is:

-x + y + z = -4 \\ -x + 5y - 23z = -32 \\ 7x - 3y - 31z = 0 \end{cases}$$ This can be written in augmented matrix form as: $$\begin{bmatrix} -1 & 1 & 1 & | & -4 \\ -1 & 5 & -23 & | & -32 \\ 7 & -3 & -31 & | & 0 \\ \end{bmatrix}$$ From here, you would perform row operations to simplify this matrix, aiming to determine if it has a unique solution, infinitely many solutions, or if it is inconsistent. Here’s a quick guide on how to proceed: 1. **Make the first entry in the first row a 1**, if possible, by scaling or swapping rows. 2. **Use row operations** to make all other entries in the first column zero. 3. **Move to the second column**, repeating similar operations to make it a leading 1 and then zeroing out other entries. 4. Continue this process to simplify the matrix fully. Would you like a detailed walkthrough of the row operations, or do you have any questions about specific steps? Here are some related questions you might find helpful: 1. How can I identify when a system is inconsistent through row operations? 2. What does it mean if I end up with a row of all zeros in the matrix? 3. How do I interpret a row with zeros except in the augmented part? 4. Can you have an inconsistent system with two rows only? 5. What are the implications of having a free variable in this context? **Tip:** Always check if scaling a row first makes it easier to simplify other rows.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations
Row Echelon Form

Formulas

Matrix representation of a system of equations
Row operations: Row swapping, scaling, and row addition/subtraction

Theorems

Row Echelon Form Theorem
Existence and Uniqueness Theorem for Linear Systems

Suitable Grade Level

Grades 10-12

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