Question

Obtain the solution set of the system x – 3y + 4z = 9, 4x +3y + 2z = 7, y – 2x = 5 – 10z by
elimination.

Answer 1

Answer on Question #45755 – Engineering – Other

Obtain the solution set of the system $x - 3y + 4z = 9$, $4x + 3y + 2z = 7$, $y - 2x = 5 - 10z$ by elimination.

Solution:

The elimination method can be used to solve a system of linear equations. By adding or subtracting the three linear equations in a way that eliminates one of the variables, a single variable equation is left

$$\left\{ \begin{array}{l l} x - 3y + 4z = 9 & (1) \\ 4x + 3y + 2z = 7 & (2) \\ y - 2x = 5 - 10z & (3) \end{array} \right.$$ $$(1) + (2):$$ $$x - 3y + 4z + (4x + 3y + 2z) = 9 + 7$$ $$5x + 6z = 16$$ $$z = \frac{16 - 5x}{6} \quad (4)$$ $$(2) \times -2:$$ $$\left\{ \begin{array}{l l} x - 3y + 4z = 9 & (1) \\ -8x - 6y - 4z = -14 & (2) \\ y - 2x + 10z = 5 & (3) \end{array} \right.$$ $$(2) + (1):$$ $$x - 3y + 4z + (-8x - 6y - 4z) = 9 - 14$$ $$-7x - 9y = -5$$ $$y = \frac{5 - 7x}{9} \quad (5)$$ $$(5) \text{ and } (4) \text{ in } (3):$$ $$\frac{5 - 7x}{9} - 2x = 5 - 10 \cdot \frac{16 - 5x}{6}$$ $$4(5 - 7x) - 72x = 180 - 60(16 - 5x)$$ $$800 - 400x = 0$$ $$y = \frac{5 - 7x}{9} = \frac{5 - 7 \cdot 2}{9} = -1$$ $$z = \frac{16 - 5x}{6} = \frac{16 - 5 \cdot 2}{6} = 1$$

Answer: $x = 2; y = -1; z = 1$;

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