Finite Math Examples

$19 = 1 x + 1 y + 1 z$ , $15 = 1 x + 2 y + 0 z$ , $13 = 1 x + 1 y + 1 z$

Step 1

Solve for $x$ in $19 = x + y + z$ .

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Step 1.1

Rewrite the equation as $x + y + z = 19$ .

$x + y + z = 19$

$15 = x + 2 y$

$13 = x + y + z$

Step 1.2

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

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Step 1.2.1

Subtract $y$ from both sides of the equation.

$x + z = 19 - y$

$15 = x + 2 y$

$13 = x + y + z$

Step 1.2.2

Subtract $z$ from both sides of the equation.

$x = 19 - y - z$

$15 = x + 2 y$

$13 = x + y + z$

$x = 19 - y - z$

$15 = x + 2 y$

$13 = x + y + z$

$x = 19 - y - z$

$15 = x + 2 y$

$13 = x + y + z$

Step 2

Replace all occurrences of $x$ with $19 - y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $15 = x + 2 y$ with $19 - y - z$ .

$15 = (19 - y - z) + 2 y$

$x = 19 - y - z$

$13 = x + y + z$

Step 2.2

Simplify the right side.

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Step 2.2.1

Add $- y$ and $2 y$ .

$15 = 19 + y - z$

$x = 19 - y - z$

$13 = x + y + z$

$15 = 19 + y - z$

$x = 19 - y - z$

$13 = x + y + z$

Step 2.3

Replace all occurrences of $x$ in $13 = x + y + z$ with $19 - y - z$ .

$13 = (19 - y - z) + y + z$

$15 = 19 + y - z$

$x = 19 - y - z$

Step 2.4

Simplify the right side.

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Step 2.4.1

Combine the opposite terms in $(19 - y - z) + y + z$ .

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Step 2.4.1.1

Add $- y$ and $y$ .

$13 = 19 - z + 0 + z$

$15 = 19 + y - z$

$x = 19 - y - z$

Step 2.4.1.2

Add $19 - z$ and $0$ .

$13 = 19 - z + z$

$15 = 19 + y - z$

$x = 19 - y - z$

Step 2.4.1.3

Add $- z$ and $z$ .

$13 = 19 + 0$

$15 = 19 + y - z$

$x = 19 - y - z$

Step 2.4.1.4

Add $19$ and $0$ .

$13 = 19$

$15 = 19 + y - z$

$x = 19 - y - z$

$13 = 19$

$15 = 19 + y - z$

$x = 19 - y - z$

$13 = 19$

$15 = 19 + y - z$

$x = 19 - y - z$

$13 = 19$

$15 = 19 + y - z$

$x = 19 - y - z$

Step 3

Since $13 = 19$ is not true, there is no solution.

No solution