Linear Algebra Examples

$- 4 x - 2 y = 12$ , $4 x + 2 y = - 8$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} - 4 & - 2 & 12 \\ 4 & 2 & - 8 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

Multiply each element of $R_{1}$ by $- \frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot - 2 & - \frac{1}{4} \cdot 12 \\ 4 & 2 & - 8 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 3 \\ 4 & 2 & - 8 \end{array}]$

Step 2.2

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 3 \\ 4 - 4 \cdot 1 & 2 - 4 (\frac{1}{2}) & - 8 - 4 \cdot - 3 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 3 \\ 0 & 0 & 4 \end{array}]$

Step 2.3

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 3$ a $1$ .

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Step 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 3 \\ \frac{0}{4} & \frac{0}{4} & \frac{4}{4} \end{array}]$

Step 2.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & - 3 \\ 0 & 0 & 1 \end{array}]$

Step 2.4

Perform the row operation $R_{1} = R_{1} + 3 R_{2}$ to make the entry at $1, 3$ a $0$ .

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Step 2.4.1

Perform the row operation $R_{1} = R_{1} + 3 R_{2}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 3 \cdot 0 & \frac{1}{2} + 3 \cdot 0 & - 3 + 3 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{array}]$

Step 3

Use the result matrix to declare the final solutions to the system of equations.

$x + \frac{y}{2} = 0$

$0 = 1$

Step 4

Subtract $\frac{y}{2}$ from both sides of the equation.

$x = - \frac{y}{2}$

$0 = 1$

Step 5

Since $0 \neq 1$ , there are no solutions.

No solution