From equation 3: \(a = 4 - 2b\). Substitute into equation 2: \(2c - 3(4 - 2b) = 7 \implies 2c - 12 + 6b = 7 \implies 2c + 6b = 19\). From equation 1: \(c = 5 - 3b\). Substitute into \(2c + 6b = 19\): \(2(5 - 3b) + 6b = 10 - 6b + 6b = 10 = 19\), which is a contradiction. Thus, no solution exists.

["Solving a System of Equations with Contradiction: Understanding No Solution in Linear Systems", "When solving systems of linear equations, a consistent solution exists only if all equations agree with one another. However, in some algebraic cases—especially when substituting substitutions—contradictions emerge, revealing that no values of the variables satisfy all conditions simultaneously. This article explores one such contradiction arising from equations involving (a), (b), and (c), demonstrating why the system admits no solution.", "### Starting Points: Two Equations Involving Variables", "We begin with:", "- Equation 1: ( a = 4 - 2b )\n- Equation 2: ( 2c - 3a = 7 )", "Substituting Equation 1 into Equation 2 gives:\n[\n2c - 3(4 - 2b) = 7\n]\nExpanding the expression:\n[\n2c - 12 + 6b = 7\n]\nSolving for (2c + 6b):\n[\n2c + 6b = 19\n]", "We now form a new equation, Equation 3, by isolating terms:\n[\n2c + 6b = 19\n]", "### Introducing a Third Equation", "From Equation 1, express (c) in terms of (b):\n[\nc = 5 - 3b\n]", "Substitute this into the earlier combined equation:\n[\n2(5 - 3b) + 6b = 10 - 6b + 6b = 10\n]\nBut this contradicts the earlier derivation, since ( 2c + 6b = 19 ), yet substitution yields ( 10 = 19 )—a clear contradiction.", "### Interpreting the Contradiction", "This inconsistency ((10 = 19)) proves that the three equations cannot all be true simultaneously:", "- Equation 1 and substitution yield (2c + 6b = 19) from Equation 2.\n- Using (c = 5 - 3b) from Equation 1 in that equation produces (2c + 6b = 10), conflicting directly.", "Such contradictions signal the system is inconsistent. There is no set of values for (a), (b), and (c) that satisfies all three equations at once.", "### Why No Solution Exists", "When substituting (c = 5 - 3b) into the transformed form of Equation 2, the equations conflict—highlighting linear dependency issues. In essence:", "- Equation 3 implies (2c + 6b = 19)\n- But geometrically, substituting (c = 5 - 3b) forces (2c + 6b = 10), independent of (b)\n- A sum equal to both 10 and 19 is impossible", "Thus, no solution exists for this system. This outcome is critical in mathematical modeling and problem-solving: recognizing contradictions allows us to identify flawed assumptions or incompatible constraints early.", "### Key Takeaways", "- Substitution during system solving must be consistent across all equations\n- Contradictions like (10 = 19) indicate no common solution\n- Understanding such contradictions strengthens problem-solving skills in algebra and applied mathematics", "If you're encountering similar contradictions in equations, double-check substitutions and ensure each equation reflects valid, compatible relationships.", "---\nKeywords: linear equations, system inconsistency, contradiction in algebra, substitution method, no solution system, solving equations step-by-step, mathematical proofs."]