Question
define System of linear equations
Answers
GPT-4o mini
A system of linear equations is a collection of two or more linear equations that share the same set of variables. The solutions to the system are the values of these variables that satisfy all of the equations simultaneously.
Mathematically, a system of linear equations can be represented in the following general form:
\[
\begin{align*}
a_1x_1 + b_1x_2 + c_1x_3 + \ldots + d_1x_n &= e_1 \\
a_2x_1 + b_2x_2 + c_2x_3 + \ldots + d_2x_n &= e_2 \\
\vdots \\
a_mx_1 + b_mx_2 + c_mx_3 + \ldots + d_mx_n &= e_m
\end{align*}
\]
where:
- \(x_1, x_2, \ldots, x_n\) are the variables,
- \(a_i, b_i, c_i, d_i\) are the coefficients of the variables in each equation,
- \(e_i\) are the constants on the right-hand side of the equations,
- \(m\) is the number of equations in the system,
- \(n\) is the number of variables.
### Types of Solutions
A system of linear equations can have:
1. **One unique solution:** Indicates that the lines (or planes, in higher dimensions) intersect at a single point.
2. **No solution:** Occurs when the lines (or planes) are parallel and never intersect.
3. **Infinitely many solutions:** Happens when the equations are dependent, meaning they represent the same line (or plane) in the solution space.
### Methods of Solving
There are various methods to solve systems of linear equations, including:
- Substitution
- Elimination
- Matrix methods (such as Gaussian elimination)
- Graphical methods
- Using determinants (Cramer's Rule)
Systems of linear equations have wide applications in fields such as engineering, economics, physics, and statistics.
Mathematically, a system of linear equations can be represented in the following general form:
\[
\begin{align*}
a_1x_1 + b_1x_2 + c_1x_3 + \ldots + d_1x_n &= e_1 \\
a_2x_1 + b_2x_2 + c_2x_3 + \ldots + d_2x_n &= e_2 \\
\vdots \\
a_mx_1 + b_mx_2 + c_mx_3 + \ldots + d_mx_n &= e_m
\end{align*}
\]
where:
- \(x_1, x_2, \ldots, x_n\) are the variables,
- \(a_i, b_i, c_i, d_i\) are the coefficients of the variables in each equation,
- \(e_i\) are the constants on the right-hand side of the equations,
- \(m\) is the number of equations in the system,
- \(n\) is the number of variables.
### Types of Solutions
A system of linear equations can have:
1. **One unique solution:** Indicates that the lines (or planes, in higher dimensions) intersect at a single point.
2. **No solution:** Occurs when the lines (or planes) are parallel and never intersect.
3. **Infinitely many solutions:** Happens when the equations are dependent, meaning they represent the same line (or plane) in the solution space.
### Methods of Solving
There are various methods to solve systems of linear equations, including:
- Substitution
- Elimination
- Matrix methods (such as Gaussian elimination)
- Graphical methods
- Using determinants (Cramer's Rule)
Systems of linear equations have wide applications in fields such as engineering, economics, physics, and statistics.