Precalculus Examples

$4 x - y + 3 z = - 1$ , $x + y + z = 2$ , $3 x + 3 y + 3 z = 5$

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.

$x + z = 2 - y$

$4 x - y + 3 z = - 1$

$3 x + 3 y + 3 z = 5$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 2 - y - z$

$4 x - y + 3 z = - 1$

$3 x + 3 y + 3 z = 5$

$x = 2 - y - z$

$4 x - y + 3 z = - 1$

$3 x + 3 y + 3 z = 5$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $4 x - y + 3 z = - 1$ with $2 - y - z$ .

$4 (2 - y - z) - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (2 - y - z) - y + 3 z$ .

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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.

$4 \cdot 2 + 4 (- y) + 4 (- z) - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $4$ by $2$ .

$8 + 4 (- y) + 4 (- z) - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2.1.1.2.2

Multiply $- 1$ by $4$ .

$8 - 4 y + 4 (- z) - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2.1.1.2.3

Multiply $- 1$ by $4$ .

$8 - 4 y - 4 z - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

$8 - 4 y - 4 z - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

$8 - 4 y - 4 z - y + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $y$ from $- 4 y$ .

$8 - 5 y - 4 z + 3 z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.2.1.2.2

Add $- 4 z$ and $3 z$ .

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$3 x + 3 y + 3 z = 5$

Step 2.3

Replace all occurrences of $x$ in $3 x + 3 y + 3 z = 5$ with $2 - y - z$ .

$3 (2 - y - z) + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

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

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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.

$3 \cdot 2 + 3 (- y) + 3 (- z) + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $2$ .

$6 + 3 (- y) + 3 (- z) + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $3$ .

$6 - 3 y + 3 (- z) + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $3$ .

$6 - 3 y - 3 z + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 - 3 y - 3 z + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 - 3 y - 3 z + 3 y + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.2

Combine the opposite terms in $6 - 3 y - 3 z + 3 y + 3 z$ .

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Step 2.4.1.2.1

Add $- 3 y$ and $3 y$ .

$6 - 3 z + 0 + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.2.2

Add $6 - 3 z$ and $0$ .

$6 - 3 z + 3 z = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.2.3

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

$6 + 0 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 2.4.1.2.4

Add $6$ and $0$ .

$6 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

$6 = 5$

$8 - 5 y - z = - 1$

$x = 2 - y - z$

Step 3

Since $6 = 5$ is not true, there is no solution.

No solution