Math Problem Statement

Solve the following linear system x + y + z = 1, 2x + 2y + 2z = 2 and 3x + 3y + 3z = 3.

Solution

Let's solve the given system of linear equations:

1. \quad & x + y + z &= 1 \\ 2. \quad & 2x + 2y + 2z &= 2 \\ 3. \quad & 3x + 3y + 3z &= 3 \end{aligned}$$ First, notice that the second and third equations are simply multiples of the first equation: - The second equation: $$2x + 2y + 2z = 2$$ can be divided by 2, resulting in $$x + y + z = 1$$. - The third equation: $$3x + 3y + 3z = 3$$ can be divided by 3, also resulting in $$x + y + z = 1$$. Thus, all three equations are equivalent to $$x + y + z = 1$$. Since the three equations are equivalent, they represent the same plane in three-dimensional space. Therefore, there are infinitely many solutions that satisfy the equation $$x + y + z = 1$$. To express the solution set, we can parametrize two of the variables in terms of the third. Let's solve for $$x$$ and $$y$$ in terms of $$z$$: $$x + y = 1 - z$$ Let $$z = t$$, where $$t$$ is a free parameter. Then: $$x = 1 - y - t$$ So the general solution is: $$\begin{aligned} x &= 1 - y - t, \\ y &= y, \\ z &= t, \end{aligned}$$ where $$y$$ and $$t$$ are free parameters. This can be written in parametric form as: $$\begin{aligned} x &= 1 - t - s, \\ y &= s, \\ z &= t, \end{aligned}$$ where $$s$$ and $$t$$ are free parameters. ### Summary: The solution set is given by $$(x, y, z) = (1 - s - t, s, t)$$, where $$s$$ and $$t$$ are real numbers. This represents an infinite number of solutions. --- Would you like more details on any part of this solution or have any specific questions? Here are some related topics you might find interesting: 1. How to determine if a system of linear equations has no solution. 2. Solving linear systems using matrix methods (e.g., Gaussian elimination). 3. Exploring the geometric interpretation of linear systems. 4. Understanding the concept of free parameters in linear systems. 5. The role of rank in determining the solution set of a linear system. 6. Exploring dependent and independent equations in a system. 7. How to solve systems of linear equations using substitution. 8. Application of linear systems in real-world problems. **Tip:** When dealing with linear systems, always check if the equations are multiples of each other. It simplifies the process and helps to identify if the system has infinite solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Parametric Equations
Systems of Equations

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12

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