Algebra Examples

$3 x + 3 y = 3$ , $x < y$

Step 1

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

$x + Z = y$

$3 x + 3 y - 3 = 0$

Step 2

Subtract $y$ from both sides of the equation.

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

Step 3

Add $3$ to both sides of the equation.

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

Step 4

Write the system of equations in matrix form.

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

Step 5

Find the reduced row echelon form.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ 0 & 6 & 0 & 3 \end{array}]$

Step 5.2

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 5.2.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 0 & 0 \\ \frac{0}{6} & \frac{6}{6} & \frac{0}{6} & \frac{3}{6} \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify $R_{1}$ .

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

Step 6

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

$x = \frac{1}{2}$

$y = \frac{1}{2}$

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