Solving an Inconsistent System of Linear Equations with 3 Variables

Math Problem Statement

   x + y + 3z = −7 y + z = 3 2x + 4y + 8z = −4 x + 2y + 4z = −4

We are given the following system of equations:

1.\ & x + y + 3z = -7 \\ 2.\ & y + z = 3 \\ 3.\ & 2x + 4y + 8z = -4 \\ 4.\ & x + 2y + 4z = -4 \\ \end{aligned}$$ ### Step 1: Simplify the system Notice that the 4th equation is a scaled-down version of the 3rd equation. In fact, dividing the 3rd equation by 2 gives: $$2x + 4y + 8z = -4 \quad \Rightarrow \quad x + 2y + 4z = -2$$ This contradicts equation 4, which says $$x + 2y + 4z = -4$$. Therefore, **the system of equations is inconsistent** and has **no solution**. Would you like details on how to verify or modify this system? --- Here are some follow-up questions to explore: 1. How can we identify inconsistent systems of equations quickly? 2. What happens if we change one of the equations in the system? 3. How does a system of equations relate to geometric representations like planes? 4. What methods can we use to solve consistent systems of equations? 5. What is the importance of row reduction in solving systems? **Tip:** When two equations are multiples of each other but contradict in value, it indicates that the system has no solution!

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Algebra
Systems of Linear Equations
Inconsistent Systems

x + y + 3z = −7
y + z = 3
2x + 4y + 8z = −4
x + 2y + 4z = −4

If two equations in a system are multiples of each other but lead to different results, the system is inconsistent.

Grades 9-12

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