Math Problem Statement

Resolve el sistema de tres ecuaciones con tres incógnitas: \[ \begin{cases} x + 2y - z = -1 \\ 2x - y + 3z = 3 \\ 5x + y + 2z = 4 \end{cases} \]

Solution

The problem is a system of three equations with three variables xx, yy, and zz. Here’s the system extracted from the image:

x + 2y - z = -1 \\ 2x - y + 3z = 3 \\ 5x + y + 2z = 4 \end{cases}$$ To solve this system, we can use methods like substitution, elimination, or matrix operations. I'll go through the solution using elimination: 1. **Step 1: Eliminate $$x$$ from two equations.** Multiply the first equation by 2 to align it with the coefficient of $$x$$ in the second equation: $$2(x + 2y - z) = 2(-1) \Rightarrow 2x + 4y - 2z = -2$$ Now we have: $$2x + 4y - 2z = -2$$ $$2x - y + 3z = 3$$ Subtract the second equation from the modified first equation to eliminate $$x$$: $$(2x + 4y - 2z) - (2x - y + 3z) = -2 - 3$$ $$5y - 5z = -5 \Rightarrow y - z = -1$$ 2. **Step 2: Eliminate $$x$$ using the first and third equations.** Multiply the first equation by 5: $$5(x + 2y - z) = 5(-1) \Rightarrow 5x + 10y - 5z = -5$$ Now we have: $$5x + 10y - 5z = -5$$ $$5x + y + 2z = 4$$ Subtract the third equation from the modified first equation to eliminate $$x$$: $$(5x + 10y - 5z) - (5x + y + 2z) = -5 - 4$$ $$9y - 7z = -9$$ 3. **Step 3: Solve the resulting system of two equations with $$y$$ and $$z$$:** We now have: $$y - z = -1$$ $$9y - 7z = -9$$ Solve the first equation for $$y$$: $$y = z - 1$$ Substitute $$y = z - 1$$ into the second equation: $$9(z - 1) - 7z = -9$$ $$9z - 9 - 7z = -9$$ $$2z = 0 \Rightarrow z = 0$$ Substitute $$z = 0$$ back into $$y = z - 1$$: $$y = 0 - 1 = -1$$ 4. **Step 4: Substitute $$y = -1$$ and $$z = 0$$ into the first equation to find $$x$$:** $$x + 2(-1) - 0 = -1$$ $$x - 2 = -1$$ $$x = 1$$ So, the solution to the system is: $$x = 1, \quad y = -1, \quad z = 0$$ Would you like further details or have any questions? Here are some related questions to deepen your understanding: 1. How can matrix methods, such as Gaussian elimination, solve this system? 2. What is the significance of having a unique solution for this system? 3. How would the solution change if one of the equations was altered? 4. Can you solve similar systems with different numbers of variables and equations? 5. What are alternative methods for solving systems of linear equations? **Tip:** Practice different methods (substitution, elimination, matrix) to find which is most efficient for each system type.

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

Formulas

Elimination and substitution methods for systems of equations

Theorems

Unique solution criteria for linear systems

Suitable Grade Level

Grades 10-12

Related Recommendation

Solving a System of Three Linear Equations Using Elimination and Substitution

Solving Systems of Linear Equations: 2x - 6y - 4z = 16, 4x + 6y + 5z = -27, -8x + 6y - z = -9

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

Solving a System of Linear Equations with Three Variables

Solving a System of Linear Equations using Elimination and Substitution