Finite Math Examples

$6 x - 5 Y = 30$ , $12 x - 10 y = 6$

Step 1

Reorder $6 x$ and $- 5 Y$ .

$- 5 Y + 6 x = 30$

$12 x - 10 y = 6$

Step 2

Write the system as a matrix.

$[\begin{array}{c} - 5 & 6 & 0 & 30 \\ 0 & 12 & - 10 & 6 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Multiply each element of $R_{1}$ by $- \frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

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Step 3.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{5} \cdot - 5 & - \frac{1}{5} \cdot 6 & - \frac{1}{5} \cdot 0 & - \frac{1}{5} \cdot 30 \\ 0 & 12 & - 10 & 6 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 & - 6 \\ 0 & 12 & - 10 & 6 \end{array}]$

Step 3.2

Multiply each element of $R_{2}$ by $\frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

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Step 3.2.1

Multiply each element of $R_{2}$ by $\frac{1}{12}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 & - 6 \\ \frac{0}{12} & \frac{12}{12} & \frac{- 10}{12} & \frac{6}{12} \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{6}{5} & 0 & - 6 \\ 0 & 1 & - \frac{5}{6} & \frac{1}{2} \end{array}]$

Step 3.3

Perform the row operation $R_{1} = R_{1} + \frac{6}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.3.1

Perform the row operation $R_{1} = R_{1} + \frac{6}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{6}{5} \cdot 0 & - \frac{6}{5} + \frac{6}{5} \cdot 1 & 0 + \frac{6}{5} (- \frac{5}{6}) & - 6 + \frac{6}{5} \cdot \frac{1}{2} \\ 0 & 1 & - \frac{5}{6} & \frac{1}{2} \end{array}]$

Step 3.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 1 & - \frac{27}{5} \\ 0 & 1 & - \frac{5}{6} & \frac{1}{2} \end{array}]$

Step 4

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

$Y - y = - \frac{27}{5}$

$x - \frac{5}{6} y = \frac{1}{2}$

Step 5

The solution is the set of ordered pairs that make the system true.

$(- \frac{27}{5} + y, \frac{1}{2} + \frac{5 y}{6}, y)$