Solve Linear System A0E456

1. **State the problem:** Solve the system of linear equations: $$-3x + 4y - 2z = 19$$ $$x + 4y + 3z = 3$$ $$6x - 2y + 8z = -22$$ 2. **Choose a method:** We will use the elimination method to solve for $x$, $y$, and $z$. 3. **Label the equations:** $$\text{(1)}\ -3x + 4y - 2z = 19$$ $$\text{(2)}\ x + 4y + 3z = 3$$ $$\text{(3)}\ 6x - 2y + 8z = -22$$ 4. **Eliminate $y$ from (1) and (2):** Subtract (2) from (1): $$(-3x + 4y - 2z) - (x + 4y + 3z) = 19 - 3$$ Simplify: $$-3x - x + 4y - 4y - 2z - 3z = 16$$ $$-4x - 5z = 16$$ This is equation (4). 5. **Eliminate $y$ from (2) and (3):** Multiply (2) by $1$ and (3) by $2$ to align $y$ coefficients: $$\text{(2)}: x + 4y + 3z = 3$$ $$2 \times \text{(3)}: 12x - 4y + 16z = -44$$ Add these equations: $$(x + 4y + 3z) + (12x - 4y + 16z) = 3 + (-44)$$ Simplify: $$x + 12x + 4y - 4y + 3z + 16z = -41$$ $$13x + 19z = -41$$ This is equation (5). 6. **Solve the system of two equations with two variables:** Equations (4) and (5): $$-4x - 5z = 16$$ $$13x + 19z = -41$$ Multiply (4) by $13$ and (5) by $4$ to eliminate $x$: $$13 \times (-4x - 5z) = 13 \times 16 \Rightarrow -52x - 65z = 208$$ $$4 \times (13x + 19z) = 4 \times (-41) \Rightarrow 52x + 76z = -164$$ Add these two equations: $$(-52x - 65z) + (52x + 76z) = 208 + (-164)$$ Simplify: $$(-52x + 52x) + (-65z + 76z) = 44$$ $$11z = 44$$ Divide both sides by 11: $$z = \cancel{\frac{11z}{11}} = \cancel{\frac{44}{11}} = 4$$ 7. **Substitute $z=4$ into equation (4):** $$-4x - 5(4) = 16$$ $$-4x - 20 = 16$$ Add 20 to both sides: $$-4x = 36$$ Divide both sides by $-4$: $$x = \cancel{\frac{-4x}{-4}} = \cancel{\frac{36}{-4}} = -9$$ 8. **Substitute $x = -9$ and $z = 4$ into equation (2) to find $y$:** $$-9 + 4y + 3(4) = 3$$ Simplify: $$-9 + 4y + 12 = 3$$ $$4y + 3 = 3$$ Subtract 3 from both sides: $$4y = 0$$ Divide both sides by 4: $$y = \cancel{\frac{4y}{4}} = \cancel{\frac{0}{4}} = 0$$ 9. **Final solution:** $$x = -9, \quad y = 0, \quad z = 4$$