Finite Math Examples

$y - 5 x = 30$ , $y > x + 2$

Step 1

Introduce slack variables $u$ and $v$ to replace the inequalities with equations.

$y - Z = x + 2$

$y - 5 x - 30 = 0$

Step 2

Subtract $x$ from both sides of the equation.

$y - Z - x = 2, y - 5 x - 30 = 0$

Step 3

Add $30$ to both sides of the equation.

$y - Z - x = 2, y - 5 x = 30$

Step 4

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & 1 & 0 & 2 \\ - 5 & 1 & 0 & 30 \end{array}]$

Step 5

Find the reduced row echelon form.

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Step 5.1

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

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Step 5.1.1

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

$[\begin{array}{c} - - 1 & - 1 \cdot 1 & - 0 & - 1 \cdot 2 \\ - 5 & 1 & 0 & 30 \end{array}]$

Step 5.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 2 \\ - 5 & 1 & 0 & 30 \end{array}]$

Step 5.2

Perform the row operation $R_{2} = R_{2} + 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.2.1

Perform the row operation $R_{2} = R_{2} + 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 2 \\ - 5 + 5 \cdot 1 & 1 + 5 \cdot - 1 & 0 + 5 \cdot 0 & 30 + 5 \cdot - 2 \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 2 \\ 0 & - 4 & 0 & 20 \end{array}]$

Step 5.3

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

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Step 5.3.1

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

$[\begin{array}{c} 1 & - 1 & 0 & - 2 \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot 20 \end{array}]$

Step 5.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & - 2 \\ 0 & 1 & 0 & - 5 \end{array}]$

Step 5.4

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

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Step 5.4.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & - 2 - 5 \\ 0 & 1 & 0 & - 5 \end{array}]$

Step 5.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 7 \\ 0 & 1 & 0 & - 5 \end{array}]$

Step 6

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

$x = 0$

$y = 0$