Solve the System of Equations 0.5x+0.25y+0.1z=4.7 , y=2z-10 , x-y=2

$0.5 x + 0.25 y + 0.1 z = 4.7$ , $y = 2 z - 10$ , $x - y = 2$

Step 1

Replace all occurrences of $y$ with $2 z - 10$ in each equation.

Replace all occurrences of $y$ in $0.5 x + 0.25 y + 0.1 z = 4.7$ with $2 z - 10$ .

$0.5 x + 0.25 (2 z - 10) + 0.1 z = 4.7$

$y = 2 z - 10$

$x - y = 2$

Simplify the left side.

Simplify $0.5 x + 0.25 (2 z - 10) + 0.1 z$ .

Simplify each term.

Apply the distributive property.

$0.5 x + 0.25 (2 z) + 0.25 \cdot - 10 + 0.1 z = 4.7$

$y = 2 z - 10$

$x - y = 2$

Multiply $2$ by $0.25$ .

$0.5 x + 0.5 z + 0.25 \cdot - 10 + 0.1 z = 4.7$

$y = 2 z - 10$

$x - y = 2$

Multiply $0.25$ by $- 10$ .

$0.5 x + 0.5 z - 2.5 + 0.1 z = 4.7$

$y = 2 z - 10$

$x - y = 2$

$0.5 x + 0.5 z - 2.5 + 0.1 z = 4.7$

$y = 2 z - 10$

$x - y = 2$

Add $0.5 z$ and $0.1 z$ .

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - y = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - y = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - y = 2$

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

$x - (2 z - 10) = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$x - (2 z) + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Multiply $2$ by $- 1$ .

$x - 2 z + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Multiply $- 1$ by $- 10$ .

$x - 2 z + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - 2 z + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - 2 z + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x - 2 z + 10 = 2$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Step 2

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

Add $2 z$ to both sides of the equation.

$x + 10 = 2 + 2 z$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Subtract $10$ from both sides of the equation.

$x = 2 + 2 z - 10$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Subtract $10$ from $2$ .

$x = 2 z - 8$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

$x = 2 z - 8$

$0.5 x + 0.6 z - 2.5 = 4.7$

$y = 2 z - 10$

Step 3

Replace all occurrences of $x$ with $2 z - 8$ in each equation.

Replace all occurrences of $x$ in $0.5 x + 0.6 z - 2.5 = 4.7$ with $2 z - 8$ .

$0.5 (2 z - 8) + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Simplify the left side.

Simplify $0.5 (2 z - 8) + 0.6 z - 2.5$ .

Simplify each term.

Apply the distributive property.

$0.5 (2 z) + 0.5 \cdot - 8 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Multiply $0.5 (2 z)$ .

Multiply $2$ by $0.5$ .

$1 z + 0.5 \cdot - 8 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Multiply $z$ by $1$ .

$z + 0.5 \cdot - 8 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$z + 0.5 \cdot - 8 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Multiply $0.5$ by $- 8$ .

$z - 4 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$z - 4 + 0.6 z - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Simplify by adding terms.

Add $z$ and $0.6 z$ .

$1.6 z - 4 - 2.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Subtract $2.5$ from $- 4$ .

$1.6 z - 6.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$1.6 z - 6.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$1.6 z - 6.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$1.6 z - 6.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

$1.6 z - 6.5 = 4.7$

$x = 2 z - 8$

$y = 2 z - 10$

Step 4

Solve for $z$ in $1.6 z - 6.5 = 4.7$ .

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

Add $6.5$ to both sides of the equation.

$1.6 z = 4.7 + 6.5$

$x = 2 z - 8$

$y = 2 z - 10$

Add $4.7$ and $6.5$ .

$1.6 z = 11.2$

$x = 2 z - 8$

$y = 2 z - 10$

$1.6 z = 11.2$

$x = 2 z - 8$

$y = 2 z - 10$

Divide each term in $1.6 z = 11.2$ by $1.6$ and simplify.

Divide each term in $1.6 z = 11.2$ by $1.6$ .

$\frac{1.6 z}{1.6} = \frac{11.2}{1.6}$

$x = 2 z - 8$

$y = 2 z - 10$

Simplify the left side.

Cancel the common factor of $1.6$ .

Cancel the common factor.

$\frac{1.6 z}{1.6} = \frac{11.2}{1.6}$

$x = 2 z - 8$

$y = 2 z - 10$

Divide $z$ by $1$ .

$z = \frac{11.2}{1.6}$

$x = 2 z - 8$

$y = 2 z - 10$

$z = \frac{11.2}{1.6}$

$x = 2 z - 8$

$y = 2 z - 10$

$z = \frac{11.2}{1.6}$

$x = 2 z - 8$

$y = 2 z - 10$

Simplify the right side.

Divide $11.2$ by $1.6$ .

$z = 7$

$x = 2 z - 8$

$y = 2 z - 10$

$z = 7$

$x = 2 z - 8$

$y = 2 z - 10$

$z = 7$

$x = 2 z - 8$

$y = 2 z - 10$

$z = 7$

$x = 2 z - 8$

$y = 2 z - 10$

Step 5

Replace all occurrences of $z$ with $7$ in each equation.

Replace all occurrences of $z$ in $x = 2 z - 8$ with $7$ .

$x = 2 (7) - 8$

$z = 7$

$y = 2 z - 10$

Simplify the right side.

Simplify $2 (7) - 8$ .

Multiply $2$ by $7$ .

$x = 14 - 8$

$z = 7$

$y = 2 z - 10$

Subtract $8$ from $14$ .

$x = 6$

$z = 7$

$y = 2 z - 10$

$x = 6$

$z = 7$

$y = 2 z - 10$

$x = 6$

$z = 7$

$y = 2 z - 10$

Replace all occurrences of $z$ in $y = 2 z - 10$ with $7$ .

$y = 2 (7) - 10$

$x = 6$

$z = 7$

Simplify the right side.

Simplify $2 (7) - 10$ .

Multiply $2$ by $7$ .

$y = 14 - 10$

$x = 6$

$z = 7$

Subtract $10$ from $14$ .

$y = 4$

$x = 6$

$z = 7$

$y = 4$

$x = 6$

$z = 7$

$y = 4$

$x = 6$

$z = 7$

$y = 4$

$x = 6$

$z = 7$

Step 6

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

$(6, 4, 7)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, 4, 7)$

Equation Form:

$x = 6, y = 4, z = 7$

Solve the System of Equations 0.5x+0.25y+0.1z=4.7 , y=2z-10 , x-y=2

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