Question

systems of equations

Answer 1

A system of equations is a set of equations with the same variable(s). The solutions of a system of equations are the values of the variables that satisfy all equations in the system simultaneously. Systems of equations can be classified as:

1. **Linear Systems**: Both equations are linear, which means they can be represented in the form \( ax + by = c \).

For example:
\[
\begin{align*}
2x + 3y &= 6 \\
4x - y &= 5
\end{align*}
\]

2. **Non-linear Systems**: At least one of the equations is non-linear.

For example:
\[
\begin{align*}
x^2 + y^2 &= 25 \\
y &= 2x + 3
\end{align*}
\]

### Methods of Solving Systems of Equations

1. **Graphical Method**: Graphs of the equations are plotted on the same coordinate plane, and the points of intersection represent the solutions.

2. **Substitution Method**: Solve one equation for one variable and substitute it into the other equation.

3. **Elimination Method**: Adjust the equations to eliminate one variable, making it possible to solve for the other.

4. **Matrix Method**: Use matrices to represent and solve the system using row reduction or the inverse of a matrix. This is particularly useful for larger systems.

### Example

Consider the following linear system:
\[
\begin{align*}
x + 2y &= 8 \quad (1) \\
3x - 4y &= -6 \quad (2)
\end{align*}
\]

**1. Graphical Method**: Plot both equations on a graph to find the intersection.

**2. Substitution Method**:
- From equation (1): \( x = 8 - 2y \)
- Substitute \( x \) in equation (2):
\[
3(8 - 2y) - 4y = -6 \\
24 - 6y = -6 \\
-6y = -30 \\
y = 5
\]
- Substitute \( y \) back into equation (1):
\[
x + 2(5) = 8 \\
x + 10 = 8 \\
x = -2
\]

**Solution**: \( x = -2, y = 5 \)

### Types of Solutions

1. **Unique Solution**: The lines (or curves) intersect at one point.
2. **No Solution**: The lines (or curves) are parallel and never intersect.
3. **Infinite Solutions**: The lines (or curves) coincide and thus intersect at infinitely many points.

If you have a specific system of equations or a particular method you want to explore, feel free to ask!