Systems of Linear Equations Calculator

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Example

Created on 2024-06-20Asked by Samuel Taylor (Solvelet student)

Solve the system of linear equations:

\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 14 \\ -x + 4y + z = -2 \end{cases}

Solution

To solve the system of linear equations:

\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 14 \\ -x + 4y + z = -2 \end{cases}

1. **Write the augmented matrix:**

\left(\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 2 & -1 & 3 & 14 \\ -1 & 4 & 1 & -2 \end{array}\right).

2. **Use Gaussian elimination:** - Multiply the first row by 2 and subtract from the second row. - Add the first row to the third row. - Multiply the new second row and subtract from the third row. 3. **Back-substitution:** - Solve for

z

. - Substitute

z

to solve for

y

. - Substitute

z

and

y

to solve for

x

. 4. **Result:** The solution is

x = 3

y = 1

z = 2

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Eleanor Ramirez on Solvelet

1. Solve the system of linear equations:

3x+2y=8

4x−y=3

.2. Determine if the system of equations has a unique solution:

2x-3y=7

4x-6y=14

DefinitionLinear equations contain two or more linear equations with the same variables. The common intersection of all the equations are the solutions. For example, the system 2x+3y=6 and x−y=2 has the solution x=3 and y=1.

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