1.1: System of Linear equations

Introduction to System of Linear equations

Under Construction!

Definition: Linear Equations

\(a_1x_1+a_2x_2+ \cdots+ a_nx_n=b\) is called a linear equation in \(n\) variables \(x_1, x_2, \cdots, x_n\), where \(a_i \in \mathbb{R}\) is the coefficient of \(x_i\), for \(i=1,\cdots , n\) and \(b \in \mathbb{R}\) is the constant term.

Example \(\PageIndex{1}\)

1. \(x=2\) lis a inear equation in one variable \(x\).
2. \(x+y=3\) is a linear equation in two variables \(x\) and \(y\).
3. \(x^2+y=3\) is not a linear equation.

Example \(\PageIndex{2}\)

Find two numbers whose sum is \(12\) and whose positive difference is \(3\).

Solution

Let \(x\) and \(y\) be numbers such that \(x+y =12\) and \(x-y=3\). This is a system of two linear equations in two variables.

Then we can solve these equations by eliminating one variable.

By adding the equations, we get \(2x=15\). Therefore, \(x= \dfrac{15}{2}=7.5\).

Now, \(y=12-x=12-7.5=4.5\).

Thus the numbers are \(7.5\) and \(4.5\). Thus the system has a unique solution.

We can also find the solution using geometry. Remember linear equations in two variables represent lines in two dimensions. From the graph in Figure \(\PageIndex{1}\), we can see that the intersection point is \((7.5,4.5)\).

Figure \(\PageIndex{1}\): Example \(\PageIndex{2}\). (Copyright; Pamini Thangarajah via Desmos)

Definitions:

1. \(a_1x_1+a_2x_2+ \cdots+ a_nx_n=0\) is called a homogeneous linear equation in \(n\) variables \(x_1, x_2, \cdots, x_n\).

2. A system of linear equations in the variables, \(x_{1},x_{2}\cdots ,x_{n}\), is a finite collection of linear equations. \begin{eqnarray*} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&b_{1} \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&b_{2} \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&b_{m} \end{eqnarray*} where \(a_{ij}\) and \(b_{j}\) are real numbers. The above is a system of \(m\) equations in the \(n\) variables, \(x_{1},x_{2}\cdots ,x_{n}\). Written more simply in terms of summation notation, the above can be written in the form \[\sum_{j=1}^{n}a_{ij}x_{j}=b_{i}, \text{ }i=1,2,3,\cdots ,m\]

3. A system of linear equations is called a homogeneous system if the constant term of each equation in the system is equal to \(0\). A homogeneous system has the form \[\begin{eqnarray*}{c} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n} &= &0 \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&= &0 \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=& 0 \end{eqnarray*}\] where \(a_{ij}\) are scalars and \(x_{i}\) are variable

4. A list \( (s_1,s_2, \cdots,s_n)\) is a solution of the system of linear equations if it satisfies all the linear equations in the system.

Example \(\PageIndex{3}\)

\[\begin{align} &x &+y &&-3z&=3 \\ &2x & -y&& &=4 \\ &4x &+2y &&+3z&=7 \end{align}\nonumber \]

Solutions to System of Linear equations

Possible solutions:

Consider the following systems of two equations in two variables.

\begin{equation} \left.
\begin{aligned} x-y=0\\x+y=0 \end{aligned}
\right\} \end{equation}

\begin{equation} \left. \begin{aligned} x-y=0\\x-y=1\end{aligned}
\right\} \end{equation}

\begin{equation} \left. \begin{aligned}x-y=0\\2x-2y=0\end{aligned}
\right\} \end{equation}

What are the solutions to the above systems.

Definition:

A system of linear equations is called consistent if there exists at least one solution. It is called inconsistent if there is no solution.

Augmented matrix

Definition: Elementary Row operations

The elementary row operations consist of the following

1. Interchange two rows.

2. Multiply a row by a nonzero number.

3. Replace a row with any multiple of another row added to it.