Math Problem Statement

Solve the system of equations: 1) x + 9y + z = 20 2) x + 10y - 2z = 18 3) 3x + 27y + 2z = 58

Solution

The system of equations given is:

1) & \quad x + 9y + z = 20 \\ 2) & \quad x + 10y - 2z = 18 \\ 3) & \quad 3x + 27y + 2z = 58 \\ \end{aligned}$$ ### Step 1: Solve using elimination or substitution We can use elimination to solve this system. #### Subtract Equation 1 from Equation 2: $$(x + 10y - 2z) - (x + 9y + z) = 18 - 20$$ $$x + 10y - 2z - x - 9y - z = -2$$ $$y - 3z = -2 \quad \text{(Equation 4)}$$ #### Subtract 3 times Equation 1 from Equation 3: $$(3x + 27y + 2z) - 3(x + 9y + z) = 58 - 3(20)$$ $$3x + 27y + 2z - 3x - 27y - 3z = 58 - 60$$ $$-z = -2 \quad \Rightarrow \quad z = 2$$ ### Step 2: Substitute $$z = 2$$ into Equation 4: $$y - 3(2) = -2$$ $$y - 6 = -2 \quad \Rightarrow \quad y = 4$$ ### Step 3: Substitute $$y = 4$$ and $$z = 2$$ into Equation 1: $$x + 9(4) + 2 = 20$$ $$x + 36 + 2 = 20 \quad \Rightarrow \quad x + 38 = 20 \quad \Rightarrow \quad x = -18$$ ### Final Solution: The solution as an ordered triple is: $$(x, y, z) = (-18, 4, 2)$$ Let me know if you'd like more details or have any further questions! --- Here are some related questions for further practice: 1. How can you verify if the solution $$(-18, 4, 2)$$ satisfies all the equations? 2. What is the geometric interpretation of a system of linear equations? 3. How does substitution compare to elimination when solving a system of equations? 4. What are some special cases that can arise when solving a system of equations (e.g., no solution, infinite solutions)? 5. How can you use matrices to solve this system? **Tip:** Always check your solution by substituting it back into all original equations to ensure correctness!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Substitution Method
Elimination Method

Formulas

Basic operations with linear equations
Elimination technique
Substitution technique

Theorems

Linear Combination Theorem
Solution Methods for Linear Systems

Suitable Grade Level

Grades 9-12

Related Recommendation

Solve the System of Linear Equations 3x + y + z = 15, 2x - 3y + z = 0, x - y - 2z = -19

Solving a System of Three Linear Equations Using Elimination and Substitution

Solving a System of Linear Equations: 5x - 2y + 5z = 83, 3x - 2y - 2z = 9, x - y + 4z = 40

Solving a System of Three Linear Equations with Three Variables

Solving a System of Linear Equations with Two Equations and Three Variables