Solving a System of Linear Equations Using Gaussian Elimination

Math Problem Statement

Solve the system of equations using the Gaussian elimination method: (15 - 15)x + y = c, 15x - 2y + z = 0, -x - 2y + z = 0.

The system of equations given in the image is:

(15 - 15)x + y = c, \\ 15x - 2y + z = 0, \\ -x - 2y + z = 0. \end{cases}$$ Simplify the first equation, since $$15 - 15 = 0$$: $$y = c.$$ Now, substitute $$y = c$$ into the other two equations: 1. $$15x - 2y + z = 0 \Rightarrow 15x - 2c + z = 0 \Rightarrow z = -15x + 2c$$, 2. $$-x - 2y + z = 0 \Rightarrow -x - 2c + z = 0 \Rightarrow z = x + 2c$$. Equating the expressions for $$z$$ from the second and third equations: $$-15x + 2c = x + 2c.$$ Simplify: $$-15x = x \implies -16x = 0 \implies x = 0.$$ Substitute $$x = 0$$ into $$z = -15x + 2c$$: $$z = -15(0) + 2c = 2c.$$ Thus, the solution to the system is: $$x = 0, \; y = c, \; z = 2c.$$ ### Final Answer: $$\boxed{x = 0, \; y = c, \; z = 2c.}$$ Would you like further explanation or verification? ### Related Questions: 1. How does the Gaussian elimination method handle systems with parameter constants (e.g., $$c$$)? 2. What are the implications of a solution containing free parameters for the geometry of the system? 3. Can this system be represented in matrix form, and what would the reduced row echelon form look like? 4. How would the solution change if a constraint is added, like $$c = 0$$? 5. How does substituting back to check solutions ensure correctness? ### Tip: When solving systems with parameters, treat the parameter as a constant during computations but verify its impact on the overall system behavior at the end.

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Linear Equations
Systems of Equations
Gaussian Elimination

Substitution of variables
Equating expressions for the same variable

Gaussian elimination method

Grades 10-12

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