Finite Math Examples

$x + 2 z = 4$ , $\frac{11}{2} x + y + 10 z = 19$ , $4 y - 4 z = - 12$

Step 1

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

$x = 4 - 2 z$

$\frac{11}{2} x + y + 10 z = 19$

$4 y - 4 z = - 12$

Step 2

Replace all occurrences of $x$ with $4 - 2 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{11}{2} x + y + 10 z = 19$ with $4 - 2 z$ .

$\frac{11}{2} \cdot (4 - 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{11}{2} \cdot (4 - 2 z) + y + 10 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$\frac{11}{2} \cdot 4 + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $4$ .

$\frac{11}{2} \cdot (2 (2)) + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.2.2

Cancel the common factor.

$\frac{11}{2} \cdot (2 \cdot 2) + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.2.3

Rewrite the expression.

$11 \cdot 2 + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$11 \cdot 2 + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.3

Multiply $11$ by $2$ .

$22 + \frac{11}{2} \cdot (- 2 z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Factor $2$ out of $- 2 z$ .

$22 + \frac{11}{2} \cdot (2 (- z)) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.4.2

Cancel the common factor.

$22 + \frac{11}{2} \cdot (2 (- z)) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.4.3

Rewrite the expression.

$22 + 11 (- z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$22 + 11 (- z) + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.1.5

Multiply $- 1$ by $11$ .

$22 - 11 z + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$22 - 11 z + y + 10 z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 2.2.1.2

Add $- 11 z$ and $10 z$ .

$22 + y - z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$22 + y - z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$22 + y - z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$22 + y - z = 19$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 3

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

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Step 3.1

Subtract $22$ from both sides of the equation.

$y - z = 19 - 22$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 3.2

Add $z$ to both sides of the equation.

$y = 19 - 22 + z$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 3.3

Subtract $22$ from $19$ .

$y = - 3 + z$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$4 y - 4 z = - 12$

Step 4

Replace all occurrences of $y$ with $- 3 + z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $4 y - 4 z = - 12$ with $- 3 + z$ .

$4 (- 3 + z) - 4 z = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 (- 3 + z) - 4 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$4 \cdot - 3 + 4 z - 4 z = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

Step 4.2.1.1.2

Multiply $4$ by $- 3$ .

$- 12 + 4 z - 4 z = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$- 12 + 4 z - 4 z = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

Step 4.2.1.2

Combine the opposite terms in $- 12 + 4 z - 4 z$ .

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Step 4.2.1.2.1

Subtract $4 z$ from $4 z$ .

$- 12 + 0 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

Step 4.2.1.2.2

Add $- 12$ and $0$ .

$- 12 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$- 12 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$- 12 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$- 12 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

$- 12 = - 12$

$y = - 3 + z$

$x = 4 - 2 z$

Step 5

Remove any equations from the system that are always true.

$y = - 3 + z$

$x = 4 - 2 z$