Math Problem Statement
Selesaikan menggunakan metode substitusi dan eliminasi
Solution
The system of equations you provided is:
1) & \quad 3x + 2y + 5z = 2 \\ 2) & \quad 5x + 3y - 2z = 4 \\ 3) & \quad 2x - 5y - 3z = 14 \end{aligned}$$ To solve this system using the substitution and elimination methods, follow these steps: ### Step 1: Elimination of one variable Let's eliminate $$z$$ by manipulating equations 1) and 2). Multiply equation 1 by 2 and equation 2 by 5, then add them together to eliminate $$z$$. $$\begin{aligned} 2 \times (3x + 2y + 5z) &= 2 \times 2 \quad \text{(Multiply equation 1 by 2)} \\ 6x + 4y + 10z &= 4 \quad \text{(Equation 4)} \end{aligned}$$ $$\begin{aligned} 5 \times (5x + 3y - 2z) &= 5 \times 4 \quad \text{(Multiply equation 2 by 5)} \\ 25x + 15y - 10z &= 20 \quad \text{(Equation 5)} \end{aligned}$$ Now, add equation 4 and equation 5: $$(6x + 4y + 10z) + (25x + 15y - 10z) = 4 + 20$$ $$31x + 19y = 24 \quad \text{(Equation 6)}$$ ### Step 2: Elimination of the same variable $$z$$ using another pair of equations Now use equation 1) and 3) to eliminate $$z$$. Multiply equation 1 by 3 and equation 3 by 5: $$\begin{aligned} 3 \times (3x + 2y + 5z) &= 3 \times 2 \quad \text{(Multiply equation 1 by 3)} \\ 9x + 6y + 15z &= 6 \quad \text{(Equation 7)} \end{aligned}$$ $$\begin{aligned} 5 \times (2x - 5y - 3z) &= 5 \times 14 \quad \text{(Multiply equation 3 by 5)} \\ 10x - 25y - 15z &= 70 \quad \text{(Equation 8)} \end{aligned}$$ Add equation 7 and equation 8: $$(9x + 6y + 15z) + (10x - 25y - 15z) = 6 + 70$$ $$19x - 19y = 76 \quad \text{(Equation 9)}$$ $$x - y = 4 \quad \text{(Equation 10)}$$ ### Step 3: Substitute and solve for $$x$$ and $$y$$ From equation 10, $$x = y + 4$$. Substitute $$x = y + 4$$ into equation 6: $$31(y + 4) + 19y = 24$$ $$31y + 124 + 19y = 24$$ $$50y = -100$$ $$y = -2$$ Substitute $$y = -2$$ back into equation 10: $$x = -2 + 4 = 2$$ ### Step 4: Solve for $$z$$ Use equation 1) to solve for $$z$$: $$3(2) + 2(-2) + 5z = 2$$ $$6 - 4 + 5z = 2$$ $$2 + 5z = 2$$ $$5z = 0$$ $$z = 0$$ ### Final Answer: The solution to the system of equations is: $$x = 2, \quad y = -2, \quad z = 0$$ Would you like more details on any step, or do you have any other questions? --- **Related Questions:** 1. How would the solution change if the third equation was modified slightly? 2. What if one of the variables had a non-zero initial condition, how would the approach differ? 3. Can substitution and elimination methods be used effectively in larger systems of equations, like 4 or 5 variables? 4. How do you solve a system of equations graphically compared to algebraically? 5. How can matrix operations be applied to solve this system of equations? **Tip:** When using the elimination method, always choose the variable to eliminate first by considering which will simplify the process most efficiently.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Linear Equations
Substitution Method
Elimination Method
Formulas
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Theorems
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Suitable Grade Level
High School
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