Solving a system of equations involves finding the values of the variables that satisfy all equations simultaneously. There are several methods to solve a system, depending on the type and number of equations. Here are the most common methods:

### 1. **Substitution Method**:
   - Solve one equation for one variable.
   - Substitute that expression into the other equation(s).
   - Solve the resulting equation for the other variable(s).
   - Back-substitute to find the value of the first variable.

   **Example**:
   Solve the system: 
   \[
   x + y = 5
   \]
   \[
   2x - y = 1
   \]

   - Solve the first equation for \( y \): \( y = 5 - x \).
   - Substitute \( y = 5 - x \) into the second equation: 
     \[
     2x - (5 - x) = 1
     \]
   - Solve for \( x \): 
     \[
     2x - 5 + x = 1 \implies 3x = 6 \implies x = 2
     \]
   - Back-substitute \( x = 2 \) into \( y = 5 - x \) to find \( y = 3 \).

   So, the solution is \( x = 2 \), \( y = 3 \).

### 2. **Elimination Method (Addition or Subtraction)**:
   - Multiply or add/subtract the equations to eliminate one variable.
   - Solve for the remaining variable.
   - Substitute back into one of the original equations to find the other variable.

   **Example**:
   Solve the system:
   \[
   3x + 2y = 16
   \]
   \[
   4x - 2y = 8
   \]

   - Add the equations: 
     \[
     (3x + 2y) + (4x - 2y) = 16 + 8 \implies 7x = 24 \implies x = \frac{24}{7}
     \]
   - Substitute \( x = \frac{24}{7} \) into one of the original equations to find \( y \).

### 3. **Matrix Method (Using Matrices)**:
   - Write the system as a matrix equation \( A \mathbf{x} = \mathbf{b} \).
   - Use matrix operations (like Gaussian elimination or the inverse of the matrix \( A \)) to solve for \( \mathbf{x} \).

   **Example**:
   Given the system:
   \[
   x + 2y = 5
   \]
   \[
   3x - y = 4
   \]
   The matrix form is:
   \[
   \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}
   \]
   You can solve this system using the inverse of the coefficient matrix or Gaussian elimination.

### 4. **Graphing Method**:
   - Graph both equations on a coordinate plane.
   - The point where the lines intersect is the solution.

   **Example**:
   Graph the system:
   \[
   y = 2x + 1
   \]
   \[
   y = -x + 3
   \]
   The intersection point of the two lines will be the solution.

### 5. **Cramer's Rule (for 2x2 and 3x3 systems)**:
   - Cramer's Rule is a method that uses determinants of matrices to solve systems of linear equations.

   **Example**:
   For a system of two equations:
   \[
   a_1x + b_1y = c_1
   \]
   \[
   a_2x + b_2y = c_2
   \]
   The solution is:
   \[
   x = \frac{\text{det}(C_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(C_y)}{\text{det}(A)}
   \]
   where \( C_x \) and \( C_y \) are matrices formed by replacing columns in the coefficient matrix \( A \) with the constants from the right-hand side of the equations.

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Would you like me to provide an example using any of these methods, or do you have any specific questions?

Here are 5 related questions:
1. How do you use substitution in non-linear systems of equations?
2. What is Gaussian elimination and how does it solve a system of equations?
3. How do you apply Cramer’s rule to a 3x3 system?
4. How can you determine if a system of equations has no solution or infinite solutions?
5. How can matrix inverses be used to solve a system of equations?

**Tip**: When graphing, if the lines are parallel, the system has no solution (inconsistent), and if the lines coincide, it has infinitely many solutions (dependent).

Learn how to solve systems of equations with methods like substitution, elimination, matrix operations, and Cramer’s Rule. Step-by-step examples are provided, making it perfect for high school students (Grades 9-12).

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