Finite Math Examples

y - 3 < 5

$y - 3 < 5$ ,

y = 9

$y = 9$

Step 1

Introduce slack variables

u

$u$ and

v

$v$ to replace the inequalities with equations.

y - 3 + Z = 5

$y - 3 + Z = 5$

y - 9 = 0

$y - 9 = 0$

Step 2

Add

3

$3$ to both sides of the equation.

y + Z = 5 + 3, y - 9 = 0

$y + Z = 5 + 3, y - 9 = 0$

Step 3

Add

5

$5$ and

3

$3$ .

y + Z = 8, y - 9 = 0

$y + Z = 8, y - 9 = 0$

Step 4

Add

9

$9$ to both sides of the equation.

y + Z = 8, y = 9

$y + Z = 8, y = 9$

Step 5

Write the system of equations in matrix form.

[\begin{matrix} 1 & 0 & 8 1 & 0 & 9 \end{matrix}]

$[\begin{array}{c} 1 & 0 & 8 \\ 1 & 0 & 9 \end{array}]$

Step 6

Find the reduced row echelon form.

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Step 6.1

Perform the row operation

R_{2} = R_{2} - R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

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Step 6.1.1

Perform the row operation

R_{2} = R_{2} - R_{1}

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

2, 1

$2, 1$ a

0

$0$ .

[\begin{matrix} 1 & 0 & 8 1 - 1 & 0 - 0 & 9 - 8 \end{matrix}]

$[\begin{array}{c} 1 & 0 & 8 \\ 1 - 1 & 0 - 0 & 9 - 8 \end{array}]$

Step 6.1.2

Simplify

R_{2}

$R_{2}$ .

[\begin{matrix} 1 & 0 & 8 0 & 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & 0 & 8 \\ 0 & 0 & 1 \end{array}]$

[\begin{matrix} 1 & 0 & 8 0 & 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 & 0 & 8 \\ 0 & 0 & 1 \end{array}]$

Step 6.2

Perform the row operation

R_{1} = R_{1} - 8 R_{2}

$R_{1} = R_{1} - 8 R_{2}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Step 6.2.1

Perform the row operation

R_{1} = R_{1} - 8 R_{2}

$R_{1} = R_{1} - 8 R_{2}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

[\begin{matrix} 1 - 8 \cdot 0 & 0 - 8 \cdot 0 & 8 - 8 \cdot 1 0 & 0 & 1 \end{matrix}]

$[\begin{array}{c} 1 - 8 \cdot 0 & 0 - 8 \cdot 0 & 8 - 8 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 6.2.2

Simplify

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Step 7

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

y = 0

$y = 0$

0 = 1

$0 = 1$

Step 8

Since

0 \neq 1

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

No solution