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Chapter 4: Problem 14

Use row operations to solve each system. $$ \begin{array}{r} {2 x-4 y=8} \\ {-3 x+6 y=5} \end{array} $$

Short Answer

Expert verified
The system has no solution.

Step by step solution

01

- Write the system in augmented matrix form

Convert the given system of equations to an augmented matrix. The system:
02

- Make the leading coefficient of the first row 1

Divide the first row by 2 so that the leading coefficient of the first row becomes 1.
03

- Eliminate the first column entry of the second row

Multiply the first row by 3 and add it to the second row to eliminate the x-term in the second row.
04

- Make the leading coefficient of the second row 1

Divide the second row by 6 so that the leading coefficient of the second row becomes 1.
05

- Express the results as equations

Rewrite the augmented matrix back into equation form to find the values of x and y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Row Operations
When solving systems of equations, row operations are essential. They simplify the matrix that represents the system. Row operations include:
  • Swapping rows: Interchanging two rows helps position useful pivots in place.
  • Multiplying a row by a scalar: This operation changes the row but keeps the system's equivalency.
  • Adding or subtracting rows: Combine rows to eliminate variables.
The goal is to use these operations systematically until the matrix is in its simplest form, ideally leading to easy-to-read solutions for variables.
Augmented Matrix
To solve a system of equations using row operations, start by converting the system into an augmented matrix. An augmented matrix combines the coefficients and constants of the system. For example, the system
\( \begin{array}{r} \text{2x - 4y = 8} ewline \text{-3x + 6y = 5} \ \end{array} \)
is written as:
\[ \left[ \begin{array}{cc|c} 2 & -4 & 8 \ -3 & 6 & 5 \end{array} \right] \]
This makes row operations straightforward. The augmented matrix aligns coefficients and constants in a clear, manipulable format, helping visually identify the operations needed to simplify the system.
Leading Coefficient
The leading coefficient is the first non-zero number in a row within a matrix. It defines the start of the row's contribution to solving the system. For example, in the row
\( 2x - 4y = 8 \), the leading coefficient is 2.
In Gaussian elimination, we aim to make this leading coefficient 1, often by dividing the entire row by the coefficient. This simplification makes further elimination steps easier. It鈥檚 a vital step in turning the matrix into a form where back-substitution can be used to find variable values.
Gaussian Elimination
Gaussian elimination involves using row operations to simplify a system of equations to its row-echelon form. This form makes back-substitution straightforward. The process involves:
  • Converting the system into an augmented matrix.
  • Making leading coefficients 1 by dividing rows by the coefficients.
  • Eliminating sub-leading coefficients by adding/subtracting multiples of rows from one another.
The goal is a matrix where each row鈥檚 leading coefficient is 1, and all entries below these leading coefficients are zero:
\[ \left[ \begin{array}{cc|c} 1 & a & b \ 0 & 1 & c \end{array} \right] \]
Once Gaussian elimination is completed, transform the matrix back into equation form to identify the values of the system's variables easily.

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