Solve the System of Equations 2x-3y+z=10 y+2z=13 z=5

$2 x - 3 y + z = 10$ $y + 2 z = 13$ $z = 5$

Step 1

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

Replace all occurrences of $z$ in $2 x - 3 y + z = 10$ with $5$ .

$2 x - 3 y + 5 = 10$

$z = 5$

$y + 2 z = 13$

Simplify the left side.

Remove parentheses.

$2 x - 3 y + 5 = 10$

$z = 5$

$y + 2 z = 13$

$2 x - 3 y + 5 = 10$

$z = 5$

$y + 2 z = 13$

Replace all occurrences of $z$ in $y + 2 z = 13$ with $5$ .

$y + 2 (5) = 13$

$2 x - 3 y + 5 = 10$

$z = 5$

Simplify the left side.

Multiply $2$ by $5$ .

$y + 10 = 13$

$2 x - 3 y + 5 = 10$

$z = 5$

$y + 10 = 13$

$2 x - 3 y + 5 = 10$

$z = 5$

$y + 10 = 13$

$2 x - 3 y + 5 = 10$

$z = 5$

Step 2

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

Subtract $10$ from both sides of the equation.

$y = 13 - 10$

$2 x - 3 y + 5 = 10$

$z = 5$

Subtract $10$ from $13$ .

$y = 3$

$2 x - 3 y + 5 = 10$

$z = 5$

$y = 3$

$2 x - 3 y + 5 = 10$

$z = 5$

Step 3

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

Replace all occurrences of $y$ in $2 x - 3 y + 5 = 10$ with $3$ .

$2 x - 3 \cdot 3 + 5 = 10$

$y = 3$

$z = 5$

Simplify the left side.

Simplify $2 x - 3 \cdot 3 + 5$ .

Multiply $- 3$ by $3$ .

$2 x - 9 + 5 = 10$

$y = 3$

$z = 5$

Add $- 9$ and $5$ .

$2 x - 4 = 10$

$y = 3$

$z = 5$

$2 x - 4 = 10$

$y = 3$

$z = 5$

$2 x - 4 = 10$

$y = 3$

$z = 5$

$2 x - 4 = 10$

$y = 3$

$z = 5$

Step 4

Solve for $x$ in $2 x - 4 = 10$ .

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

Add $4$ to both sides of the equation.

$2 x = 10 + 4$

$y = 3$

$z = 5$

Add $10$ and $4$ .

$2 x = 14$

$y = 3$

$z = 5$

$2 x = 14$

$y = 3$

$z = 5$

Divide each term in $2 x = 14$ by $2$ and simplify.

Divide each term in $2 x = 14$ by $2$ .

$\frac{2 x}{2} = \frac{14}{2}$

$y = 3$

$z = 5$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{14}{2}$

$y = 3$

$z = 5$

Divide $x$ by $1$ .

$x = \frac{14}{2}$

$y = 3$

$z = 5$

$x = \frac{14}{2}$

$y = 3$

$z = 5$

$x = \frac{14}{2}$

$y = 3$

$z = 5$

Simplify the right side.

Divide $14$ by $2$ .

$x = 7$

$y = 3$

$z = 5$

$x = 7$

$y = 3$

$z = 5$

$x = 7$

$y = 3$

$z = 5$

$x = 7$

$y = 3$

$z = 5$

Step 5

Solve the system of equations $x = 7 y = 3 z = 5$ .

$x = 7 y = 3 z = 5$

Step 6

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

$(7, 3, 5)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(7, 3, 5)$

Equation Form:

$x = 7, y = 3, z = 5$

Solve the System of Equations 2x-3y+z=10 y+2z=13 z=5

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