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Cramer’s Rule to solve for the letter(s) indicated within brackets in each of
following systems of equations:


Ask by Cook Lewis. in South Africa
Mar 15,2025

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To solve the given systems of equations using Cramer’s Rule, we need to calculate the determinants of the coefficient matrices and the corresponding matrices where the constants replace the columns. Here’s a simplified summary of the steps for each problem:
  1. Problem 1: Solve for
    • System of Equations:
    • Steps:
      • Calculate the determinant of the coefficient matrix .
      • Replace the second column of with the constants to find .
      • Use Cramer’s Rule: .
  2. Problem 2: Solve for
    • System of Equations:
    • Steps:
      • Rearrange equations to standard form.
      • Calculate and by replacing the third column with constants.
      • Apply Cramer’s Rule: .
  3. Problem 3: Solve for
    • System of Equations:
    • Steps:
      • Write the system in matrix form.
      • Calculate and by replacing the second column with constants.
      • Use Cramer’s Rule: .
  4. Problem 4: Solve for
    • System of Equations:
    • Steps:
      • Rearrange equations to standard form.
      • Calculate and by replacing the second column with constants.
      • Apply Cramer’s Rule: .
Summary:
For each system, set up the coefficient matrix and the matrix by replacing the column corresponding to the variable you’re solving for with the constants. Calculate the determinants and , then apply Cramer’s Rule to find the value of the variable.

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Bonus Knowledge

Cramer’s Rule is a handy method for solving systems of linear equations using determinants! The determinant of the coefficient matrix plays a central role, where if the determinant isn’t zero, unique solutions exist. Just remember, each respective variable , , or gets its own modified matrix where the column corresponding to the variable is replaced by the constants from the equations. It’s like a mathematical dance party where each variable gets its moment to shine!
For real-world applications, Cramer’s Rule can come in handy in fields like engineering, economics, and physics, where systems of equations frequently model complex problems. Imagine calculating currents in a circuit or optimizing resource allocation—Cramer’s Rule helps find solutions quickly. However, for large systems, it might become cumbersome, so consider using matrix methods for efficiency. It’s like choosing between a fun game of twister or a quick jog—both are great, but one may save you time when you’re in a hurry!

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