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:
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: .
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: .
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: .
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!