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.
Solution
To solve the given problems using Cramer’s Rule, we will first extract the systems of equations and then set up the corresponding matrices. We will then calculate the determinants needed for Cramer’s Rule.
Problem 1: Solve for
The system of equations is:
Rearranging the second and third equations:
(from the second equation)
(from the third equation)
Now, we can write the system in matrix form :
[
]
Next, we will calculate the determinant of the coefficient matrix and the determinant of the matrix (where the second column is replaced by the constants).
Step 1: Calculate
Step 2: Calculate
Replace the second column of with the constants:
Now, let’s calculate both determinants.
Problem 1 Determinants Calculation
We will calculate both determinants in parallel.
Problem 2: Solve for
The system of equations is:
Rearranging the equations:
(from the first equation)
(from the second equation)
(from the third equation)
Now, we can write the system in matrix form:
[
]
Step 1: Calculate
Step 2: Calculate
Replace the third column of with the constants:
Problem 2 Determinants Calculation
We will calculate both determinants in parallel.
Problem 3: Solve for
The system of equations is:
Rearranging the equations:
Now, we can write the system in matrix form:
[
]
Step 1: Calculate
Step 2: Calculate
Replace the second column of with the constants:
Problem 3 Determinants Calculation
We will calculate both determinants in parallel.
Problem 4: Solve for
The system of equations is:
Rearranging the equations:
Now, we can write the system in matrix form:
[
]
Step 1: Calculate
[
|A| = \
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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!