Math Problem Statement
Solution
The problem asks to find all values of for which the system has no solution. Let's focus on exercise 36 in the image:
The system of equations given is:
1. \quad x_1 + 2x_2 &= -3 \\ 2. \quad ax_1 - 2x_2 &= 5 \end{aligned}$$ To determine the values of $$a$$ that make the system inconsistent (i.e., it has no solution), we can approach this by examining the system in terms of linear dependence between the two equations. ### Step 1: Elimination Method We aim to eliminate one variable. Multiply the first equation by $$a$$ to align the $$x_1$$ terms: $$a(x_1 + 2x_2) = a(-3)$$ $$ax_1 + 2ax_2 = -3a$$ Now, subtract the second equation from this new equation: $$(ax_1 + 2ax_2) - (ax_1 - 2x_2) = -3a - 5$$ Simplifying: $$2ax_2 + 2x_2 = -3a - 5$$ $$(2a + 2)x_2 = -3a - 5$$ ### Step 2: Determine when there is no solution For the system to have no solution, the value of $$x_2$$ must be inconsistent. This happens if the left side is zero but the right side is not. Therefore, set the coefficient of $$x_2$$ to zero: $$2a + 2 = 0$$ $$a = -1$$ ### Step 3: Substitute $$a = -1$$ Substitute $$a = -1$$ into the second equation to check if the system becomes inconsistent: $$-3x_2 = -3(-1) - 5 = 3 - 5 = -2$$ This simplifies to: $$-4x_2 = -2$$ which gives $$x_2 = \frac{1}{2}$$. Thus, the system is consistent when $$a = -1$$. So, the only value of $$a$$ that leads to no solution is when the right-hand side and the left-hand side contradict. Therefore, there are no specific conditions leading to contradictions for $$a = -1$$. Would you like more details or have any questions? Here are five related questions: 1. How would you solve the system if $$a = 0$$? 2. What does it mean graphically if a system has no solution? 3. How do you determine if a system of linear equations is inconsistent? 4. How can you solve the system using matrix methods? 5. What changes in the system when $$a$$ takes positive or negative values? **Tip:** When solving systems of equations, always check for cases where variables cancel out, leading to contradictions or infinitely many solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Inconsistency in Systems
Formulas
Substitution and elimination methods
Criteria for inconsistent systems (parallel lines, no intersection)
Theorems
Inconsistency in Linear Systems
No solution condition for systems of equations
Suitable Grade Level
Grades 10-12
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