$$  $$How can you verify the solution of a system of linear equations?


To verify the solution of a system of linear equations, you can follow these steps:

### 1. **Substitute the Solution into the Original Equations**
   - Take each variable value from the solution and substitute it back into the original equations.
   - Perform the arithmetic to see if each equation is satisfied (i.e., the left-hand side equals the right-hand side).

### 2. **Check Each Equation**
   - If the solution is correct, substituting the values back into each equation should yield true statements. If any equation is not satisfied, the solution is incorrect.

### 3. **Use Matrix Multiplication (if solved using matrices)**
   - If you used a matrix method like $ AX = B $, where $ A $ is the coefficient matrix, $ X $ is the solution vector, and $ B $ is the constants vector:
     - Multiply the matrix $ A $ by the solution vector $ X $.
     - The result should be the same as the original constants vector $ B $.

### Example Verification:
Given the system of equations:
\[
\begin{aligned}
2I_1 - I_2 + 3I_3 + 4I_4 & = 9 \\
I_1 - 2I_3 + 7I_4 & = 11 \\
3I_1 - 3I_2 + I_3 + 5I_4 & = 8 \\
2I_1 + I_2 + 4I_3 + 4I_4 & = 10 \\
\end{aligned}
\]
With the solution $ I_1 = -1 $, $ I_2 = 0 $, $ I_3 = 1 $, $ I_4 = 2 $, substitute these values into each equation:

1. **For the first equation:**
   \[
   2(-1) - 0 + 3(1) + 4(2) = -2 + 3 + 8 = 9 \quad (\text{True})
   \]

2. **For the second equation:**
   \[
   -1 - 2(1) + 7(2) = -1 - 2 + 14 = 11 \quad (\text{True})
   \]

3. **For the third equation:**
   \[
   3(-1) - 3(0) + 1(1) + 5(2) = -3 + 1 + 10 = 8 \quad (\text{True})
   \]

4. **For the fourth equation:**
   \[
   2(-1) + 0 + 4(1) + 4(2) = -2 + 4 + 8 = 10 \quad (\text{True})
   \]

Since all equations are satisfied, the solution is verified as correct.

### Related Questions to Explore:
1. What are the common errors when substituting solutions back into equations?
2. How can you use determinant properties to verify the uniqueness of a solution?
3. What are the implications of rounding errors in numerical solutions of linear equations?
4. Can a system of linear equations be verified graphically? If so, how?
5. How does the condition number of a matrix affect the reliability of solutions?

**Tip:** Always recheck your arithmetic when verifying solutions manually to avoid simple calculation mistakes!

Learn how to verify the solution of a system of linear equations using substitution and matrix multiplication methods. Includes detailed examples and common verification techniques. Suitable for high school students and anyone interested in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation