Math Problem Statement

Solve the following system of equations for all three variables: -2x - 3y + 5z = -6 x + 2y - 5z = 3 -4x + y - 5z = 8

Solution

The system of equations provided is:

1. & \quad -2x - 3y + 5z = -6 \\ 2. & \quad x + 2y - 5z = 3 \\ 3. & \quad -4x + y - 5z = 8 \\ \end{aligned}$$ We will solve this system of linear equations using either substitution or elimination. Let's proceed step by step: ### Step 1: Solve the second equation for $$x$$ From equation (2): $$x + 2y - 5z = 3$$ Solve for $$x$$: $$x = 3 - 2y + 5z \quad \text{(Equation A)}$$ ### Step 2: Substitute $$x$$ into the other two equations Substitute $$x = 3 - 2y + 5z$$ from Equation A into equations (1) and (3). #### Substitution in equation (1): $$-2(3 - 2y + 5z) - 3y + 5z = -6$$ Simplifying: $$-6 + 4y - 10z - 3y + 5z = -6$$ $$y - 5z = 0 \quad \text{(Equation B)}$$ #### Substitution in equation (3): $$-4(3 - 2y + 5z) + y - 5z = 8$$ Simplifying: $$-12 + 8y - 20z + y - 5z = 8$$ $$9y - 25z = 20 \quad \text{(Equation C)}$$ ### Step 3: Solve the system of two equations (B and C) Now solve the system: $$\begin{aligned} B: & \quad y - 5z = 0 \\ C: & \quad 9y - 25z = 20 \\ \end{aligned}$$ From equation $$B$$, solve for $$y$$: $$y = 5z$$ ### Step 4: Substitute $$y = 5z$$ into equation $$C$$ Substitute $$y = 5z$$ into equation $$C$$: $$9(5z) - 25z = 20$$ Simplify: $$45z - 25z = 20$$ $$20z = 20$$ $$z = 1$$ ### Step 5: Solve for $$y$$ and $$x$$ Now that $$z = 1$$, substitute it back into the expression for $$y$$: $$y = 5(1) = 5$$ Finally, substitute $$y = 5$$ and $$z = 1$$ into Equation A to find $$x$$: $$x = 3 - 2(5) + 5(1)$$ $$x = 3 - 10 + 5 = -2$$ ### Solution The solution to the system is: $$x = -2, \quad y = 5, \quad z = 1$$ Would you like further details or have any questions? --- Here are some additional related questions: 1. How can we verify the solution by substituting it back into the original equations? 2. What is the geometric interpretation of the solution of this system of equations? 3. Can this system be solved using matrix methods like Gaussian elimination? 4. How does substitution compare to elimination for solving systems of equations? 5. What happens if one of the equations is inconsistent or dependent? **Tip:** Always verify your solutions by plugging them back into the original system to ensure they satisfy all the equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

Substitution method
Elimination method

Theorems

Fundamental Theorem of Algebra
Solving Systems of Linear Equations

Suitable Grade Level

Grades 9-12

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