Finite Math Examples

$1 x + 1 y + 1 z = 7$ , $1 x - 1 y + 2 z = 7$ , $2 x + 3 z = 15$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 7 - y$

$x - 1 y + 2 z = 7$

$2 x + 3 z = 15$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 7 - y - z$

$x - 1 y + 2 z = 7$

$2 x + 3 z = 15$

$x = 7 - y - z$

$x - 1 y + 2 z = 7$

$2 x + 3 z = 15$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x - 1 y + 2 z = 7$ with $7 - y - z$ .

$(7 - y - z) - 1 y + 2 z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(7 - y - z) - 1 y + 2 z$ .

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Step 2.2.1.1

Rewrite $- 1 y$ as $- y$ .

$7 - y - z - y + 2 z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

Step 2.2.1.2

Subtract $y$ from $- y$ .

$7 - 2 y - z + 2 z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

Step 2.2.1.3

Add $- z$ and $2 z$ .

$7 - 2 y + z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$2 x + 3 z = 15$

Step 2.3

Replace all occurrences of $x$ in $2 x + 3 z = 15$ with $7 - y - z$ .

$2 (7 - y - z) + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (7 - y - z) + 3 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$2 \cdot 7 + 2 (- y) + 2 (- z) + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $7$ .

$14 + 2 (- y) + 2 (- z) + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $2$ .

$14 - 2 y + 2 (- z) + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$14 - 2 y - 2 z + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$14 - 2 y - 2 z + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$14 - 2 y - 2 z + 3 z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 2.4.1.2

Add $- 2 z$ and $3 z$ .

$14 - 2 y + z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$14 - 2 y + z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$14 - 2 y + z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$14 - 2 y + z = 15$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 3

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

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Step 3.1

Subtract $14$ from both sides of the equation.

$- 2 y + z = 15 - 14$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 3.2

Add $2 y$ to both sides of the equation.

$z = 15 - 14 + 2 y$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 3.3

Subtract $14$ from $15$ .

$z = 1 + 2 y$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$z = 1 + 2 y$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Step 4

Replace all occurrences of $z$ with $1 + 2 y$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $7 - 2 y + z = 7$ with $1 + 2 y$ .

$7 - 2 y + 1 + 2 y = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $7 - 2 y + 1 + 2 y$ .

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Step 4.2.1.1

Remove parentheses.

$7 - 2 y + 1 + 2 y = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

Step 4.2.1.2

Combine the opposite terms in $7 - 2 y + 1 + 2 y$ .

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Step 4.2.1.2.1

Add $- 2 y$ and $2 y$ .

$7 + 0 + 1 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

Step 4.2.1.2.2

Add $7$ and $0$ .

$7 + 1 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

$7 + 1 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

Step 4.2.1.3

Add $7$ and $1$ .

$8 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

$8 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

$8 = 7$

$z = 1 + 2 y$

$x = 7 - y - z$

Step 4.3

Replace all occurrences of $z$ in $x = 7 - y - z$ with $1 + 2 y$ .

$x = 7 - y - (1 + 2 y)$

$8 = 7$

$z = 1 + 2 y$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $7 - y - (1 + 2 y)$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = 7 - y - 1 \cdot 1 - (2 y)$

$8 = 7$

$z = 1 + 2 y$

Step 4.4.1.1.2

Multiply $- 1$ by $1$ .

$x = 7 - y - 1 - (2 y)$

$8 = 7$

$z = 1 + 2 y$

Step 4.4.1.1.3

Multiply $2$ by $- 1$ .

$x = 7 - y - 1 - 2 y$

$8 = 7$

$z = 1 + 2 y$

$x = 7 - y - 1 - 2 y$

$8 = 7$

$z = 1 + 2 y$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $1$ from $7$ .

$x = - y + 6 - 2 y$

$8 = 7$

$z = 1 + 2 y$

Step 4.4.1.2.2

Subtract $2 y$ from $- y$ .

$x = - 3 y + 6$

$8 = 7$

$z = 1 + 2 y$

$x = - 3 y + 6$

$8 = 7$

$z = 1 + 2 y$

$x = - 3 y + 6$

$8 = 7$

$z = 1 + 2 y$

$x = - 3 y + 6$

$8 = 7$

$z = 1 + 2 y$

$x = - 3 y + 6$

$8 = 7$

$z = 1 + 2 y$

Step 5

Since $8 = 7$ is not true, there is no solution.

No solution