Algebra Examples

$2 x - y - z = 4$ $- x + 2 y + z = 1$ $3 x + y + z = 16$

Step 1

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

Tap for more steps...

Step 1.1

Add $x$ to both sides of the equation.

$2 y + z = 1 + x$

$2 x - y - z = 4$

$3 x + y + z = 16$

Step 1.2

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

$z = 1 + x - 2 y$

$2 x - y - z = 4$

$3 x + y + z = 16$

$z = 1 + x - 2 y$

$2 x - y - z = 4$

$3 x + y + z = 16$

Step 2

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

Tap for more steps...

Step 2.1

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

$2 x - y - (1 + x - 2 y) = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Apply the distributive property.

$2 x - y - 1 \cdot 1 - x - (- 2 y) = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.2.1.1.2

Simplify.

Tap for more steps...

Step 2.2.1.1.2.1

Multiply $- 1$ by $1$ .

$2 x - y - 1 - x - (- 2 y) = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 1$ .

$2 x - y - 1 - x + 2 y = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

$2 x - y - 1 - x + 2 y = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

$2 x - y - 1 - x + 2 y = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1.2.1

Subtract $x$ from $2 x$ .

$x - y - 1 + 2 y = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.2.1.2.2

Add $- y$ and $2 y$ .

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$3 x + y + z = 16$

Step 2.3

Replace all occurrences of $z$ in $3 x + y + z = 16$ with $1 + x - 2 y$ .

$3 x + y + 1 + x - 2 y = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

Step 2.4

Simplify the left side.

Tap for more steps...

Step 2.4.1

Simplify $3 x + y + 1 + x - 2 y$ .

Tap for more steps...

Step 2.4.1.1

Remove parentheses.

$3 x + y + 1 + x - 2 y = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

Step 2.4.1.2

Add $3 x$ and $x$ .

$4 x + y + 1 - 2 y = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

Step 2.4.1.3

Subtract $2 y$ from $y$ .

$4 x - y + 1 = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$4 x - y + 1 = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$4 x - y + 1 = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

$4 x - y + 1 = 16$

$x + y - 1 = 4$

$z = 1 + x - 2 y$

Step 3

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

Tap for more steps...

Step 3.1

Subtract $y$ from both sides of the equation.

$x - 1 = 4 - y$

$4 x - y + 1 = 16$

$z = 1 + x - 2 y$

Step 3.2

Add $1$ to both sides of the equation.

$x = 4 - y + 1$

$4 x - y + 1 = 16$

$z = 1 + x - 2 y$

Step 3.3

Add $4$ and $1$ .

$x = - y + 5$

$4 x - y + 1 = 16$

$z = 1 + x - 2 y$

$x = - y + 5$

$4 x - y + 1 = 16$

$z = 1 + x - 2 y$

Step 4

Replace all occurrences of $x$ with $- y + 5$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $4 x - y + 1 = 16$ with $- y + 5$ .

$4 (- y + 5) - y + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify $4 (- y + 5) - y + 1$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Apply the distributive property.

$4 (- y) + 4 \cdot 5 - y + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.2.1.1.2

Multiply $- 1$ by $4$ .

$- 4 y + 4 \cdot 5 - y + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.2.1.1.3

Multiply $4$ by $5$ .

$- 4 y + 20 - y + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

$- 4 y + 20 - y + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 4.2.1.2.1

Subtract $y$ from $- 4 y$ .

$- 5 y + 20 + 1 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.2.1.2.2

Add $20$ and $1$ .

$- 5 y + 21 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = 1 + x - 2 y$

Step 4.3

Replace all occurrences of $x$ in $z = 1 + x - 2 y$ with $- y + 5$ .

$z = 1 - y + 5 - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

Step 4.4

Simplify $z = 1 - y + 5 - 2 y$ .

Tap for more steps...

Step 4.4.1

Simplify the left side.

Tap for more steps...

Step 4.4.1.1

Remove parentheses.

$z = 1 - y + 5 - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = 1 - y + 5 - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

Step 4.4.2

Simplify the right side.

Tap for more steps...

Step 4.4.2.1

Simplify $1 - y + 5 - 2 y$ .

Tap for more steps...

Step 4.4.2.1.1

Add $1$ and $5$ .

$z = - y + 6 - 2 y$

$- 5 y + 21 = 16$

$x = - y + 5$

Step 4.4.2.1.2

Subtract $2 y$ from $- y$ .

$z = - 3 y + 6$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = - 3 y + 6$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = - 3 y + 6$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = - 3 y + 6$

$- 5 y + 21 = 16$

$x = - y + 5$

$z = - 3 y + 6$

$- 5 y + 21 = 16$

$x = - y + 5$

Step 5

Solve for $y$ in $- 5 y + 21 = 16$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Subtract $21$ from both sides of the equation.

$- 5 y = 16 - 21$

$z = - 3 y + 6$

$x = - y + 5$

Step 5.1.2

Subtract $21$ from $16$ .

$- 5 y = - 5$

$z = - 3 y + 6$

$x = - y + 5$

$- 5 y = - 5$

$z = - 3 y + 6$

$x = - y + 5$

Step 5.2

Divide each term in $- 5 y = - 5$ by $- 5$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $- 5 y = - 5$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 5}{- 5}$

$z = - 3 y + 6$

$x = - y + 5$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 5}{- 5}$

$z = - 3 y + 6$

$x = - y + 5$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{- 5}$

$z = - 3 y + 6$

$x = - y + 5$

$y = \frac{- 5}{- 5}$

$z = - 3 y + 6$

$x = - y + 5$

$y = \frac{- 5}{- 5}$

$z = - 3 y + 6$

$x = - y + 5$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Divide $- 5$ by $- 5$ .

$y = 1$

$z = - 3 y + 6$

$x = - y + 5$

$y = 1$

$z = - 3 y + 6$

$x = - y + 5$

$y = 1$

$z = - 3 y + 6$

$x = - y + 5$

$y = 1$

$z = - 3 y + 6$

$x = - y + 5$

Step 6

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

Tap for more steps...

Step 6.1

Replace all occurrences of $y$ in $z = - 3 y + 6$ with $1$ .

$z = - 3 \cdot 1 + 6$

$y = 1$

$x = - y + 5$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Simplify $- 3 \cdot 1 + 6$ .

Tap for more steps...

Step 6.2.1.1

Multiply $- 3$ by $1$ .

$z = - 3 + 6$

$y = 1$

$x = - y + 5$

Step 6.2.1.2

Add $- 3$ and $6$ .

$z = 3$

$y = 1$

$x = - y + 5$

$z = 3$

$y = 1$

$x = - y + 5$

$z = 3$

$y = 1$

$x = - y + 5$

Step 6.3

Replace all occurrences of $y$ in $x = - y + 5$ with $1$ .

$x = - (1) + 5$

$z = 3$

$y = 1$

Step 6.4

Simplify the right side.

Tap for more steps...

Step 6.4.1

Simplify $- (1) + 5$ .

Tap for more steps...

Step 6.4.1.1

Multiply $- 1$ by $1$ .

$x = - 1 + 5$

$z = 3$

$y = 1$

Step 6.4.1.2

Add $- 1$ and $5$ .

$x = 4$

$z = 3$

$y = 1$

$x = 4$

$z = 3$

$y = 1$

$x = 4$

$z = 3$

$y = 1$

$x = 4$

$z = 3$

$y = 1$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4, 1, 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(4, 1, 3)$

Equation Form:

$x = 4, y = 1, z = 3$