Solve the System of Equations x+2y+z=9 x-y+3z=13 2z=10

$x + 2 y + z = 9$ $x - y + 3 z = 13$ $2 z = 10$

Step 1

Divide each term in $2 z = 10$ by $2$ and simplify.

Divide each term in $2 z = 10$ by $2$ .

$\frac{2 z}{2} = \frac{10}{2}$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{10}{2}$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

Divide $z$ by $1$ .

$z = \frac{10}{2}$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

$z = \frac{10}{2}$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

$z = \frac{10}{2}$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

Simplify the right side.

Divide $10$ by $2$ .

$z = 5$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

$z = 5$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

$z = 5$

$x + 2 y + z = 9$

$x - y + 3 z = 13$

Step 2

Replace all occurrences of $z$ with $5$ in each equation.

Replace all occurrences of $z$ in $x + 2 y + z = 9$ with $5$ .

$x + 2 y + 5 = 9$

$z = 5$

$x - y + 3 z = 13$

Simplify the left side.

Remove parentheses.

$x + 2 y + 5 = 9$

$z = 5$

$x - y + 3 z = 13$

$x + 2 y + 5 = 9$

$z = 5$

$x - y + 3 z = 13$

Replace all occurrences of $z$ in $x - y + 3 z = 13$ with $5$ .

$x - y + 3 (5) = 13$

$x + 2 y + 5 = 9$

$z = 5$

Simplify the left side.

Multiply $3$ by $5$ .

$x - y + 15 = 13$

$x + 2 y + 5 = 9$

$z = 5$

$x - y + 15 = 13$

$x + 2 y + 5 = 9$

$z = 5$

$x - y + 15 = 13$

$x + 2 y + 5 = 9$

$z = 5$

Step 3

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

Add $y$ to both sides of the equation.

$x + 15 = 13 + y$

$x + 2 y + 5 = 9$

$z = 5$

Subtract $15$ from both sides of the equation.

$x = 13 + y - 15$

$x + 2 y + 5 = 9$

$z = 5$

Subtract $15$ from $13$ .

$x = y - 2$

$x + 2 y + 5 = 9$

$z = 5$

$x = y - 2$

$x + 2 y + 5 = 9$

$z = 5$

Step 4

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

Replace all occurrences of $x$ in $x + 2 y + 5 = 9$ with $y - 2$ .

$(y - 2) + 2 y + 5 = 9$

$x = y - 2$

$z = 5$

Simplify the left side.

Simplify $(y - 2) + 2 y + 5$ .

Add $y$ and $2 y$ .

$3 y - 2 + 5 = 9$

$x = y - 2$

$z = 5$

Add $- 2$ and $5$ .

$3 y + 3 = 9$

$x = y - 2$

$z = 5$

$3 y + 3 = 9$

$x = y - 2$

$z = 5$

$3 y + 3 = 9$

$x = y - 2$

$z = 5$

$3 y + 3 = 9$

$x = y - 2$

$z = 5$

Step 5

Solve for $y$ in $3 y + 3 = 9$ .

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

Subtract $3$ from both sides of the equation.

$3 y = 9 - 3$

$x = y - 2$

$z = 5$

Subtract $3$ from $9$ .

$3 y = 6$

$x = y - 2$

$z = 5$

$3 y = 6$

$x = y - 2$

$z = 5$

Divide each term in $3 y = 6$ by $3$ and simplify.

Divide each term in $3 y = 6$ by $3$ .

$\frac{3 y}{3} = \frac{6}{3}$

$x = y - 2$

$z = 5$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{6}{3}$

$x = y - 2$

$z = 5$

Divide $y$ by $1$ .

$y = \frac{6}{3}$

$x = y - 2$

$z = 5$

$y = \frac{6}{3}$

$x = y - 2$

$z = 5$

$y = \frac{6}{3}$

$x = y - 2$

$z = 5$

Simplify the right side.

Divide $6$ by $3$ .

$y = 2$

$x = y - 2$

$z = 5$

$y = 2$

$x = y - 2$

$z = 5$

$y = 2$

$x = y - 2$

$z = 5$

$y = 2$

$x = y - 2$

$z = 5$

Step 6

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

Replace all occurrences of $y$ in $x = y - 2$ with $2$ .

$x = (2) - 2$

$y = 2$

$z = 5$

Simplify the right side.

Subtract $2$ from $2$ .

$x = 0$

$y = 2$

$z = 5$

$x = 0$

$y = 2$

$z = 5$

$x = 0$

$y = 2$

$z = 5$

Step 7

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

$(0, 2, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(0, 2, 5)$

Equation Form:

$x = 0, y = 2, z = 5$

Solve the System of Equations x+2y+z=9 x-y+3z=13 2z=10

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