Math Problem Statement

Tentukan himpunan penyelesaian dari persamaan linear berikut: 2x - 4y + z = -10, 3x + y + 2z = 60, -2x + y - z = 20.

Solution

The system of linear equations from the uploaded image is:

1) & \quad 2x - 4y + z = -10 \\ 2) & \quad 3x + y + 2z = 60 \\ 3) & \quad -2x + y - z = -20 \end{aligned}$$ We will solve this system step-by-step using **substitution or elimination**. --- ### Step 1: Simplify where possible From Equation (3): $$-2x + y - z = -20 \implies y = 2x + z - 20 \quad \text{(Rearrange for $$y$$)}$$ We'll substitute this expression for $$y$$ in the other two equations. --- ### Step 2: Substitute $$y = 2x + z - 20$$ into the first two equations **Substitute into Equation (1):** $$2x - 4(2x + z - 20) + z = -10$$ Simplify: $$2x - 8x - 4z + 80 + z = -10$$ $$-6x - 3z = -90 \implies 2x + z = 30 \quad \text{(Divide by $$-3$$)}$$ So, Equation (1) simplifies to: $$2x + z = 30 \quad \text{(Let's call this Eq (4))}$$ **Substitute into Equation (2):** $$3x + (2x + z - 20) + 2z = 60$$ Simplify: $$3x + 2x + z - 20 + 2z = 60$$ $$5x + 3z = 80 \quad \text{(Let's call this Eq (5))}$$ --- ### Step 3: Solve the two new equations We now have: 1. $$2x + z = 30$$ \quad (Eq (4)) 2. $$5x + 3z = 80$$ \quad (Eq (5)) **Solve Eq (4) for $$z$$:** $$z = 30 - 2x$$ **Substitute into Eq (5):** $$5x + 3(30 - 2x) = 80$$ Simplify: $$5x + 90 - 6x = 80$$ $$-x = -10 \implies x = 10$$ --- ### Step 4: Find $$z$$ and $$y$$ Using $$x = 10$$ in Eq (4): $$2(10) + z = 30 \implies 20 + z = 30 \implies z = 10$$ Now, substitute $$x = 10$$ and $$z = 10$$ into the expression for $$y$$: $$y = 2(10) + 10 - 20 = 20 + 10 - 20 = 10$$ --- ### Step 5: Solution The solution to the system of equations is: $$x = 10, \quad y = 10, \quad z = 10$$ --- Do you need further details or have any questions? --- Here are 5 related questions to extend your understanding: 1. What are the different methods to solve a system of linear equations? 2. How can we determine if a system of equations has no solution, one solution, or infinitely many solutions? 3. What is the determinant of a system, and how does it relate to the existence of solutions? 4. Can you solve the system using matrix operations such as Gaussian elimination? 5. How would the solution change if one of the equations was altered slightly? **Tip:** Substitution works best when one equation is already solved for a variable, while elimination is useful when you need to eliminate terms quickly.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Algebra

Formulas

Substitution method for solving systems of equations
Elimination method for solving systems of equations

Theorems

Substitution Theorem
Linear System Consistency Theorem

Suitable Grade Level

Grades 10-12

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