Algebra Examples

x + 5 y = 4 x

$x + 5 y = 4 x$ ,

x > y

$x > y$

Step 1

Introduce slack variables

u

$u$ and

v

$v$ to replace the inequalities with equations.

x - Z = y

$x - Z = y$

- 3 x + 5 y = 0

$- 3 x + 5 y = 0$

Step 2

Subtract

y

$y$ from both sides of the equation.

x - Z - y = 0, - 3 x + 5 y = 0

$x - Z - y = 0, - 3 x + 5 y = 0$

Step 3

Write the system of equations in matrix form.

[\begin{matrix} 1 & - 1 & 0 & 0 - 3 & 5 & 0 & 0 \end{matrix}]

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ - 3 & 5 & 0 & 0 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Perform the row operation

R_{2} = R_{2} + 3 R_{1}

$R_{2} = R_{2} + 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 4.1.1

Perform the row operation

R_{2} = R_{2} + 3 R_{1}

$R_{2} = R_{2} + 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{matrix} 1 & - 1 & 0 & 0 - 3 + 3 \cdot 1 & 5 + 3 \cdot - 1 & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 \end{matrix}]

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ - 3 + 3 \cdot 1 & 5 + 3 \cdot - 1 & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 \end{array}]$

Step 4.1.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & - 1 & 0 & 0 0 & 2 & 0 & 0 \end{matrix}]

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

[\begin{matrix} 1 & - 1 & 0 & 0 0 & 2 & 0 & 0 \end{matrix}]

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

Step 4.2

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 4.2.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{matrix} 1 & - 1 & 0 & 0 \frac{0}{2} & \frac{2}{2} & \frac{0}{2} & \frac{0}{2} \end{matrix}]

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

Step 4.2.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & - 1 & 0 & 0 0 & 1 & 0 & 0 \end{matrix}]

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

[\begin{matrix} 1 & - 1 & 0 & 0 0 & 1 & 0 & 0 \end{matrix}]

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

Step 4.3

Perform the row operation

R_{1} = R_{1} + R_{2}

$R_{1} = R_{1} + R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 4.3.1

Perform the row operation

R_{1} = R_{1} + R_{2}

$R_{1} = R_{1} + R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{matrix} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 0 & 1 & 0 & 0 \end{matrix}]

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

Step 4.3.2

Simplify

R_{1}

$R_{1}$ .

[\begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 \end{matrix}]

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

[\begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 \end{matrix}]

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

[\begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 \end{matrix}]

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

Step 5

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

x = 0

$x = 0$

y = 0

$y = 0$

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