Solve a System of Equations using Elimination Method: Example Problem

To solve the system of equations using the elimination method, we aim to eliminate one variable at a time by combining the equations. The system provided is:

(1) \quad 10x + 7y - 2z &= 46 \\ (2) \quad 3x - 2y + 9z &= 22 \\ (3) \quad 5x + y - 3z &= 28 \end{aligned}$$ ### Step 1: Eliminate $$z$$ To eliminate $$z$$, we will first manipulate the equations to cancel out the $$z$$ term. #### Multiply equation (3) by 2: $$10x + 2y - 6z = 56 \quad \text{(4)}$$ Now, subtract equation (4) from equation (1): $$(10x + 7y - 2z) - (10x + 2y - 6z) = 46 - 56$$ Simplifying: $$5y + 4z = -10 \quad \text{(5)}$$ ### Step 2: Eliminate $$z$$ using equation (2) and equation (3) Multiply equation (3) by 3: $$15x + 3y - 9z = 84 \quad \text{(6)}$$ Now, subtract equation (6) from equation (2): $$(3x - 2y + 9z) - (15x + 3y - 9z) = 22 - 84$$ Simplifying: $$-12x - 5y + 18z = -62 \quad \text{(7)}$$ ### Step 3: Solve the reduced system Now, use the simplified equations (5) and (7) to solve for $$x$$ and $$y$$. We first eliminate $$z$$ from these simplified equations by solving the pair: $$\begin{aligned} 5y + 4z &= -10 \quad \text{(5)} \\ -12x - 5y + 18z &= -62 \quad \text{(7)} \end{aligned}$$ From equation (5): $$4z = -10 - 5y$$ $$z = \frac{-10 - 5y}{4} = -\frac{5}{2} - \frac{5y}{4}$$ Substitute this value of $$z$$ into equation (7) and solve for $$x$$ and $$y$$. Then, substitute back to find $$z$$. I will calculate these steps now.It seems there was an issue when solving for $$z$$. I'll resolve it manually by proceeding step-by-step with the elimination and substitution process. Let's continue: ### Simplified System We have: $$z = -\frac{5}{2} - \frac{5y}{4}$$ Substitute this expression for $$z$$ into equation (7) to eliminate $$z$$ and solve for $$x$$ and $$y$$: $$-12x - 5y + 18\left(-\frac{5}{2} - \frac{5y}{4}\right) = -62$$ Expanding and simplifying, we can solve for $$x$$ and $$y$$. Let's calculate the substitution step.It seems the approach using direct substitution into the equations is leading to issues in symbolic computation. To avoid further complications, let's return to the elimination process manually and solve the reduced system directly: ### Step 1: Substitute for $$z$$ We already derived: $$z = -\frac{5}{2} - \frac{5y}{4}$$ ### Step 2: Substitute into equation (7) Substitute $$z$$ into the equation: $$-12x - 5y + 18\left(-\frac{5}{2} - \frac{5y}{4}\right) = -62$$ Let's simplify this step manually: 1. Multiply the 18 with $$-\frac{5}{2}$$ and $$-\frac{5y}{4}$$: $$18 \times -\frac{5}{2} = -45$$ $$18 \times -\frac{5y}{4} = -\frac{90y}{4} = -\frac{45y}{2}$$ 2. Substitute back: $$-12x - 5y - 45 - \frac{45y}{2} = -62$$ 3. Combine like terms and solve for $$x$$ and $$y$$. I'll simplify and solve this final expression for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can continue solving the problem manually using the steps provided earlier, or you can ask me for a simpler calculation or a different question.

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