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).