Finite Math Examples

$x + y + z + u + v = 41$ , $y - z + u - v = - 1$ , $z + u + v = 24$ , $u - v = 1$ , $v = 7$

Step 1

Replace all occurrences of $v$ with $7$ in each equation.

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Step 1.1

Replace all occurrences of $v$ in $x + y + z + u + v = 41$ with $7$ .

$x + y + z + u + 7 = 41$

$v = 7$

$y - z + u - v = - 1$

$z + u + v = 24$

$u - v = 1$

Step 1.2

Simplify the left side.

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Step 1.2.1

Remove parentheses.

$x + y + z + u + 7 = 41$

$v = 7$

$y - z + u - v = - 1$

$z + u + v = 24$

$u - v = 1$

$x + y + z + u + 7 = 41$

$v = 7$

$y - z + u - v = - 1$

$z + u + v = 24$

$u - v = 1$

Step 1.3

Replace all occurrences of $v$ in $y - z + u - v = - 1$ with $7$ .

$y - z + u - (7) = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$z + u + v = 24$

$u - v = 1$

Step 1.4

Simplify the left side.

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Step 1.4.1

Multiply $- 1$ by $7$ .

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$z + u + v = 24$

$u - v = 1$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$z + u + v = 24$

$u - v = 1$

Step 1.5

Replace all occurrences of $v$ in $z + u + v = 24$ with $7$ .

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u - v = 1$

Step 1.6

Simplify the left side.

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Step 1.6.1

Remove parentheses.

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u - v = 1$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u - v = 1$

Step 1.7

Replace all occurrences of $v$ in $u - v = 1$ with $7$ .

$u - (7) = 1$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 1.8

Simplify the left side.

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Step 1.8.1

Multiply $- 1$ by $7$ .

$u - 7 = 1$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u - 7 = 1$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u - 7 = 1$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 2

Move all terms not containing $u$ to the right side of the equation.

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Step 2.1

Add $7$ to both sides of the equation.

$u = 1 + 7$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 2.2

Add $1$ and $7$ .

$u = 8$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$u = 8$

$z + u + 7 = 24$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3

Replace all occurrences of $u$ with $8$ in each equation.

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Step 3.1

Replace all occurrences of $u$ in $z + u + 7 = 24$ with $8$ .

$z + 8 + 7 = 24$

$u = 8$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $z + 8 + 7$ .

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Step 3.2.1.1

Remove parentheses.

$z + 8 + 7 = 24$

$u = 8$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.2.1.2

Add $8$ and $7$ .

$z + 15 = 24$

$u = 8$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$z + 15 = 24$

$u = 8$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

$z + 15 = 24$

$u = 8$

$y - z + u - 7 = - 1$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.3

Replace all occurrences of $u$ in $y - z + u - 7 = - 1$ with $8$ .

$y - z + 8 - 7 = - 1$

$z + 15 = 24$

$u = 8$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify $y - z + 8 - 7$ .

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Step 3.4.1.1

Remove parentheses.

$y - z + 8 - 7 = - 1$

$z + 15 = 24$

$u = 8$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.4.1.2

Subtract $7$ from $8$ .

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$x + y + z + u + 7 = 41$

$v = 7$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$x + y + z + u + 7 = 41$

$v = 7$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$x + y + z + u + 7 = 41$

$v = 7$

Step 3.5

Replace all occurrences of $u$ in $x + y + z + u + 7 = 41$ with $8$ .

$x + y + z + 8 + 7 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

Step 3.6

Simplify the left side.

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Step 3.6.1

Simplify $x + y + z + 8 + 7$ .

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Step 3.6.1.1

Remove parentheses.

$x + y + z + 8 + 7 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

Step 3.6.1.2

Add $8$ and $7$ .

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$z + 15 = 24$

$u = 8$

$v = 7$

Step 4

Move all terms not containing $z$ to the right side of the equation.

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Step 4.1

Subtract $15$ from both sides of the equation.

$z = 24 - 15$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

Step 4.2

Subtract $15$ from $24$ .

$z = 9$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

$z = 9$

$x + y + z + 15 = 41$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

Step 5

Replace all occurrences of $z$ with $9$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $x + y + z + 15 = 41$ with $9$ .

$x + y + 9 + 15 = 41$

$z = 9$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $x + y + 9 + 15$ .

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Step 5.2.1.1

Remove parentheses.

$x + y + 9 + 15 = 41$

$z = 9$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

Step 5.2.1.2

Add $9$ and $15$ .

$x + y + 24 = 41$

$z = 9$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

$x + y + 24 = 41$

$z = 9$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

$x + y + 24 = 41$

$z = 9$

$y - z + 1 = - 1$

$u = 8$

$v = 7$

Step 5.3

Replace all occurrences of $z$ in $y - z + 1 = - 1$ with $9$ .

$y - (9) + 1 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $y - (9) + 1$ .

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Step 5.4.1.1

Multiply $- 1$ by $9$ .

$y - 9 + 1 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

Step 5.4.1.2

Add $- 9$ and $1$ .

$y - 8 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

$y - 8 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

$y - 8 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

$y - 8 = - 1$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

Step 6

Move all terms not containing $y$ to the right side of the equation.

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Step 6.1

Add $8$ to both sides of the equation.

$y = - 1 + 8$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

Step 6.2

Add $- 1$ and $8$ .

$y = 7$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

$y = 7$

$x + y + 24 = 41$

$z = 9$

$u = 8$

$v = 7$

Step 7

Replace all occurrences of $y$ with $7$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x + y + 24 = 41$ with $7$ .

$x + 7 + 24 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $x + 7 + 24$ .

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Step 7.2.1.1

Remove parentheses.

$x + 7 + 24 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

Step 7.2.1.2

Add $7$ and $24$ .

$x + 31 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

$x + 31 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

$x + 31 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

$x + 31 = 41$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

Step 8

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1

Subtract $31$ from both sides of the equation.

$x = 41 - 31$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

Step 8.2

Subtract $31$ from $41$ .

$x = 10$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

$x = 10$

$y = 7$

$z = 9$

$u = 8$

$v = 7$

Step 9

Solve the system of equations.

$x = 10 y = 7 z = 9 u = 8 v = 7$

Step 10

List all of the solutions.

$x = 10, y = 7, z = 9, u = 8, v = 7$