Finite Math Examples

$1800 = 6 x + 2 y + z$ , $1400 = 4 x + 4 y + 2 z$ , $700 = 3 x + 3 y + 3 z$

Step 1

Move variables to the left and constant terms to the right.

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Step 1.1

Move all terms containing variables to the left side of the equation.

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Step 1.1.1

Subtract $6 x$ from both sides of the equation.

$1800 - 6 x = 2 y + z$

$1400 = 4 x + 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.1.2

Subtract $2 y$ from both sides of the equation.

$1800 - 6 x - 2 y = z$

$1400 = 4 x + 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.1.3

Subtract $z$ from both sides of the equation.

$1800 - 6 x - 2 y - z = 0$

$1400 = 4 x + 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

$1800 - 6 x - 2 y - z = 0$

$1400 = 4 x + 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.2

Subtract $1800$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$1400 = 4 x + 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.3

Move all terms containing variables to the left side of the equation.

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Step 1.3.1

Subtract $4 x$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$1400 - 4 x = 4 y + 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.3.2

Subtract $4 y$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$1400 - 4 x - 4 y = 2 z$

$700 = 3 x + 3 y + 3 z$

Step 1.3.3

Subtract $2 z$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$1400 - 4 x - 4 y - 2 z = 0$

$700 = 3 x + 3 y + 3 z$

$- 6 x - 2 y - z = - 1800$

$1400 - 4 x - 4 y - 2 z = 0$

$700 = 3 x + 3 y + 3 z$

Step 1.4

Subtract $1400$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$700 = 3 x + 3 y + 3 z$

Step 1.5

Move all terms containing variables to the left side of the equation.

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Step 1.5.1

Subtract $3 x$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$700 - 3 x = 3 y + 3 z$

Step 1.5.2

Subtract $3 y$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$700 - 3 x - 3 y = 3 z$

Step 1.5.3

Subtract $3 z$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$700 - 3 x - 3 y - 3 z = 0$

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$700 - 3 x - 3 y - 3 z = 0$

Step 1.6

Subtract $700$ from both sides of the equation.

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$- 3 x - 3 y - 3 z = - 700$

$- 6 x - 2 y - z = - 1800$

$- 4 x - 4 y - 2 z = - 1400$

$- 3 x - 3 y - 3 z = - 700$

Step 2

Write the system as a matrix.

$[\begin{array}{c} - 6 & - 2 & - 1 & - 1800 \\ - 4 & - 4 & - 2 & - 1400 \\ - 3 & - 3 & - 3 & - 700 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

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

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Step 3.1.1

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

$[\begin{array}{c} - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 2 & - \frac{1}{6} \cdot - 1 & - \frac{1}{6} \cdot - 1800 \\ - 4 & - 4 & - 2 & - 1400 \\ - 3 & - 3 & - 3 & - 700 \end{array}]$

Step 3.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ - 4 & - 4 & - 2 & - 1400 \\ - 3 & - 3 & - 3 & - 700 \end{array}]$

Step 3.2

Perform the row operation $R_{2} = R_{2} + 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.2.1

Perform the row operation $R_{2} = R_{2} + 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ - 4 + 4 \cdot 1 & - 4 + 4 (\frac{1}{3}) & - 2 + 4 (\frac{1}{6}) & - 1400 + 4 \cdot 300 \\ - 3 & - 3 & - 3 & - 700 \end{array}]$

Step 3.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & - \frac{8}{3} & - \frac{4}{3} & - 200 \\ - 3 & - 3 & - 3 & - 700 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & - \frac{8}{3} & - \frac{4}{3} & - 200 \\ - 3 + 3 \cdot 1 & - 3 + 3 (\frac{1}{3}) & - 3 + 3 (\frac{1}{6}) & - 700 + 3 \cdot 300 \end{array}]$

Step 3.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & - \frac{8}{3} & - \frac{4}{3} & - 200 \\ 0 & - 2 & - \frac{5}{2} & 200 \end{array}]$

Step 3.4

Multiply each element of $R_{2}$ by $- \frac{3}{8}$ to make the entry at $2, 2$ a $1$ .

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Step 3.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{8}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ - \frac{3}{8} \cdot 0 & - \frac{3}{8} (- \frac{8}{3}) & - \frac{3}{8} (- \frac{4}{3}) & - \frac{3}{8} \cdot - 200 \\ 0 & - 2 & - \frac{5}{2} & 200 \end{array}]$

Step 3.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & \frac{1}{2} & 75 \\ 0 & - 2 & - \frac{5}{2} & 200 \end{array}]$

Step 3.5

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

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Step 3.5.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & \frac{1}{2} & 75 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - \frac{5}{2} + 2 (\frac{1}{2}) & 200 + 2 \cdot 75 \end{array}]$

Step 3.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & \frac{1}{2} & 75 \\ 0 & 0 & - \frac{3}{2} & 350 \end{array}]$

Step 3.6

Multiply each element of $R_{3}$ by $- \frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

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Step 3.6.1

Multiply each element of $R_{3}$ by $- \frac{2}{3}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & \frac{1}{2} & 75 \\ - \frac{2}{3} \cdot 0 & - \frac{2}{3} \cdot 0 & - \frac{2}{3} (- \frac{3}{2}) & - \frac{2}{3} \cdot 350 \end{array}]$

Step 3.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & \frac{1}{2} & 75 \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.7

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

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Step 3.7.1

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

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 - \frac{1}{2} \cdot 0 & 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 75 - \frac{1}{2} (- \frac{700}{3}) \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.7.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{1}{6} & 300 \\ 0 & 1 & 0 & \frac{575}{3} \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.8

Perform the row operation $R_{1} = R_{1} - \frac{1}{6} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 3.8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{6} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{6} \cdot 0 & \frac{1}{3} - \frac{1}{6} \cdot 0 & \frac{1}{6} - \frac{1}{6} \cdot 1 & 300 - \frac{1}{6} (- \frac{700}{3}) \\ 0 & 1 & 0 & \frac{575}{3} \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 & \frac{3050}{9} \\ 0 & 1 & 0 & \frac{575}{3} \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.9

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

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Step 3.9.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & 0 - \frac{1}{3} \cdot 0 & \frac{3050}{9} - \frac{1}{3} \cdot \frac{575}{3} \\ 0 & 1 & 0 & \frac{575}{3} \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 3.9.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 275 \\ 0 & 1 & 0 & \frac{575}{3} \\ 0 & 0 & 1 & - \frac{700}{3} \end{array}]$

Step 4

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

$x = 275$

$y = \frac{575}{3}$

$z = - \frac{700}{3}$

Step 5

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

$(275, \frac{575}{3}, - \frac{700}{3})$