Row Operations Calculator

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Example

Created on 2024-06-20Asked by Aiden Baker (Solvelet student)

Perform row operations to solve the system of linear equations:

\begin{cases} 2x + 3y = 5 \\ 4x + y = 6 \end{cases}

Solution

To solve the system of linear equations using row operations:

\begin{cases} 2x + 3y = 5 \\ 4x + y = 6 \end{cases}

1. **Write the augmented matrix:**

\left[\begin{array}{cc|c} 2 & 3 & 5 \\ 4 & 1 & 6 \end{array}\right].

2. **Perform row operations:** - Subtract

2 \times

(row 1) from (row 2):

R2 \leftarrow R2 - 2R1 \implies \left[\begin{array}{cc|c} 2 & 3 & 5 \\ 0 & -5 & -4 \end{array}\right].

- Divide row 2 by

-5

R2 \leftarrow \frac{R2}{-5} \implies \left[\begin{array}{cc|c} 2 & 3 & 5 \\ 0 & 1 & \frac{4}{5} \end{array}\right].

- Subtract

3 \times

(row 2) from (row 1):

R1 \leftarrow R1 - 3R2 \implies \left[\begin{array}{cc|c} 2 & 0 & \frac{7}{5} \\ 0 & 1 & \frac{4}{5} \end{array}\right].

- Divide row 1 by 2:

R1 \leftarrow \frac{R1}{2} \implies \left[\begin{array}{cc|c} 1 & 0 & \frac{7}{10} \\ 0 & 1 & \frac{4}{5} \end{array}\right].

3. **Solution:**

x = \frac{7}{10}, \quad y = \frac{4}{5}.

Therefore, the solution to the system is

x = \frac{7}{10}

and

y = \frac{4}{5}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Sofia Hernandez on Solvelet

1. Use row operations to solve the system of equations:

2x + y = 3

4x - 2y = 2

.2. Perform row reduction on the augmented matrix

\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

to obtain its row echelon form.,

DefinitionManipulations of rows of matrices to solve linear systems and find matrix inverses are known as row operations. The operations are row swapping, multiplication of a row by a non-zero scalar; and adding or subtracting multiples of rows. Example: Solving a system via Gaussian elimination involves performing row operations to convert the augmented matrix to row echelon form.

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