Substitute \( (1, 0) \): \( 0 = a(1-2)^2 - 3 \).

["Understanding the Substitution ( (1, 0) ) in the Elimination ( 0 = a(1-2)^2 - 3 ): Step-by-Step Explanation", "When solving systems of equations using elimination, substitutions like ( (1, 0) ) play a key role in simplifying and solving the system. This article explores the substitution ( (1, 0) ) applied to the equation ( 0 = a(1-2)^2 - 3 ), explaining how it supports mathematical manipulation and solution finding.", "---", "### What is the Substitution ( (1, 0) )?", "In the context of substitution within linear algebra or systems of equations, ( (1, 0) ) typically refers to substituting specific values—here, ( x = 1 ), ( y = 0 )—into an expression or equation to evaluate or simplify it. However, in the expression ( 0 = a(1-2)^2 - 3 ), ( (1, 0) ) implies assigning particular values to the unknowns in place of variables before solving for ( a ).", "---", "### Analyzing the Equation: ( 0 = a(1-2)^2 - 3 )", "Start by interpreting the equation:", "[\n0 = a(1 - 2)^2 - 3\n]", "1. Simplify the squared term:\n[\n(1 - 2)^2 = (-1)^2 = 1\n]", "2. Plug into the equation:\n[\n0 = a(1) - 3\n]", "3. This simplifies to:\n[\n0 = a - 3\n]", "4. Solving for ( a ):\n[\na = 3\n]", "---", "### How Substitution ( (1, 0) ) Supports This Solution", "While ( (1, 0) ) does not directly appear in the expression (as those are variables in a generic system), substituting ( x = 1 ), ( y = 0 ) forms the basis of plugging values into equations derived from linear systems. In broader contexts—such as verifying solutions—substituting these numbers confirms whether the derived equation holds true when ( a = 3 ).", "For instance, plugging ( a = 3 ), ( x = 1 ), ( y = 0 ) into the original system (if derived from coordinates or constraints) verifies consistency:", "[\n0 = 3(1 - 2)^2 - 3 = 3(1) - 3 = 0\n]", "Thus, substitution ensures the formulation aligns with expected results.", "---", "### Why This Substitution Matters in Elimination Methods", "In Gaussian elimination or row-reduction strategies, such substitutions help:", "- Verify validity of intermediate steps\n- Simplify expressions before substituting values\n- Confirm solutions satisfy all imposed constraints\n- Guide numerical computations by reducing symbolic complexity", "The pair ( (1, 0) )—though abstract here—reflects plugging known values to check or derive correctness in equation manipulation.", "---", "### Summary", "- The equation ( 0 = a(1 - 2)^2 - 3 ) simplifies to ( a = 3 ) by solving step-by-step.\n- Substitution ( (1, 0) ) symbolizes assigning values to variables, useful in validating solutions.\n- Such substitutions ensure algebraic expressions are correctly evaluated and aligned with system requirements.\n- Understanding substitution and simplification strengthens problem-solving in linear algebra and equation solving.", "---", "Keywords: Substitute (1, 0), equation solving, ( 0 = a(1-2)^2 - 3 ), elimination method, linear algebra substitution, verify solution, simplify algebra, solve for ( a )", "---", "Meta Description: Learn how substitution ( (1, 0) ) supports solving ( 0 = a(1-2)^2 - 3 ) by simplifying coefficients and verifying solutions in linear systems. Step-by-step explanation with algebraic clarity."]