Finite Math Examples

$1 x + 1 y + 1 z + 1 w = 7$ , $1 w + 1 x + 1 y - 1 z = 3$ , $2 x + 3 y + 1 z + 1 w = 19$ , $3 w - 1 x + 1 y + 2 z = 16$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $x$ by $1$ .

$x + 1 y + 1 z + 1 w = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.1.2

Multiply $y$ by $1$ .

$x + y + 1 z + 1 w = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.1.3

Multiply $z$ by $1$ .

$x + y + z + 1 w = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.1.4

Multiply $w$ by $1$ .

$x + y + z + w = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

$x + y + z + w = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.2

Move $z$ .

$x + y + w + z = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.3

Move $y$ .

$x + w + y + z = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.4

Reorder $x$ and $w$ .

$w + x + y + z = 7$

$1 w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.5

Simplify each term.

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Step 1.5.1

Multiply $w$ by $1$ .

$w + x + y + z = 7$

$w + 1 x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.5.2

Multiply $x$ by $1$ .

$w + x + y + z = 7$

$w + x + 1 y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.5.3

Multiply $y$ by $1$ .

$w + x + y + z = 7$

$w + x + y - 1 z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.5.4

Rewrite $- 1 z$ as $- z$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + 1 z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.6

Simplify each term.

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Step 1.6.1

Multiply $z$ by $1$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + z + 1 w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.6.2

Multiply $w$ by $1$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + z + w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + z + w = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.7

Move $z$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + 3 y + w + z = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.8

Move $3 y$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$2 x + w + 3 y + z = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.9

Reorder $2 x$ and $w$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$w + 2 x + 3 y + z = 19$

$3 w - 1 x + 1 y + 2 z = 16$

Step 1.10

Simplify each term.

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Step 1.10.1

Rewrite $- 1 x$ as $- x$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$w + 2 x + 3 y + z = 19$

$3 w - x + 1 y + 2 z = 16$

Step 1.10.2

Multiply $y$ by $1$ .

$w + x + y + z = 7$

$w + x + y - z = 3$

$w + 2 x + 3 y + z = 19$

$3 w - x + y + 2 z = 16$

$w + x + y + z = 7$

$w + x + y - z = 3$

$w + 2 x + 3 y + z = 19$

$3 w - x + y + 2 z = 16$

$w + x + y + z = 7$

$w + x + y - z = 3$

$w + 2 x + 3 y + z = 19$

$3 w - x + y + 2 z = 16$

Step 2

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 1 & 1 & 1 & - 1 & 3 \\ 1 & 2 & 3 & 1 & 19 \\ 3 & - 1 & 1 & 2 & 16 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

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

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Step 3.1.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 1 - 1 & 1 - 1 & 1 - 1 & - 1 - 1 & 3 - 7 \\ 1 & 2 & 3 & 1 & 19 \\ 3 & - 1 & 1 & 2 & 16 \end{array}]$

Step 3.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 0 & 0 & - 2 & - 4 \\ 1 & 2 & 3 & 1 & 19 \\ 3 & - 1 & 1 & 2 & 16 \end{array}]$

Step 3.2

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

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Step 3.2.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 0 & 0 & - 2 & - 4 \\ 1 - 1 & 2 - 1 & 3 - 1 & 1 - 1 & 19 - 7 \\ 3 & - 1 & 1 & 2 & 16 \end{array}]$

Step 3.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 & 1 & 2 & 0 & 12 \\ 3 & - 1 & 1 & 2 & 16 \end{array}]$

Step 3.3

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 3.3.1

Perform the row operation $R_{4} = R_{4} - 3 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 & 1 & 2 & 0 & 12 \\ 3 - 3 \cdot 1 & - 1 - 3 \cdot 1 & 1 - 3 \cdot 1 & 2 - 3 \cdot 1 & 16 - 3 \cdot 7 \end{array}]$

Step 3.3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & - 4 & - 2 & - 1 & - 5 \end{array}]$

Step 3.4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 & - 4 & - 2 & - 1 & - 5 \end{array}]$

Step 3.5

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

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Step 3.5.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 + 4 \cdot 0 & - 4 + 4 \cdot 1 & - 2 + 4 \cdot 2 & - 1 + 4 \cdot 0 & - 5 + 4 \cdot 12 \end{array}]$

Step 3.5.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 0 & - 2 & - 4 \\ 0 & 0 & 6 & - 1 & 43 \end{array}]$

Step 3.6

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 6 & - 1 & 43 \\ 0 & 0 & 0 & - 2 & - 4 \end{array}]$

Step 3.7

Multiply each element of $R_{3}$ by $\frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

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Step 3.7.1

Multiply each element of $R_{3}$ by $\frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ \frac{0}{6} & \frac{0}{6} & \frac{6}{6} & \frac{- 1}{6} & \frac{43}{6} \\ 0 & 0 & 0 & - 2 & - 4 \end{array}]$

Step 3.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & - \frac{1}{6} & \frac{43}{6} \\ 0 & 0 & 0 & - 2 & - 4 \end{array}]$

Step 3.8

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

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Step 3.8.1

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & - \frac{1}{6} & \frac{43}{6} \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 4 \end{array}]$

Step 3.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & - \frac{1}{6} & \frac{43}{6} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.9

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.9.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{6} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 + \frac{1}{6} \cdot 0 & 0 + \frac{1}{6} \cdot 0 & 1 + \frac{1}{6} \cdot 0 & - \frac{1}{6} + \frac{1}{6} \cdot 1 & \frac{43}{6} + \frac{1}{6} \cdot 2 \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.9.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.10

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

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Step 3.10.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 0 & 1 - 1 & 7 - 2 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.10.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 2 & 0 & 12 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.11

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

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Step 3.11.1

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

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 12 - 2 (\frac{15}{2}) \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.11.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 0 & 5 \\ 0 & 1 & 0 & 0 & - 3 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.12

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

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Step 3.12.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 0 - 0 & 5 - \frac{15}{2} \\ 0 & 1 & 0 & 0 & - 3 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.12.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 0 & - \frac{5}{2} \\ 0 & 1 & 0 & 0 & - 3 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.13

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

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Step 3.13.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 0 - 0 & - \frac{5}{2} + 3 \\ 0 & 1 & 0 & 0 & - 3 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 3.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0 & - 3 \\ 0 & 0 & 1 & 0 & \frac{15}{2} \\ 0 & 0 & 0 & 1 & 2 \end{array}]$

Step 4

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

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

$x = - 3$

$y = \frac{15}{2}$

$z = 2$

Step 5

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

$(\frac{1}{2}, - 3, \frac{15}{2}, 2)$