Definition. A linear equation is an equation that can be written in the form \begin{equation} a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b. \end{equation} The numbers \( a_1, a_2, \ldots, a_n \) are the coefficients of the linear equation, and \( x_1, x_2, \ldots x_n \) are the variables.
Example 1. Consider the equation \( 2x + y - 8z = 10 \). This equation is linear, with coefficients \( 2 \), \( 1 \), and \( -8 \). The variables are \( x \), \( y \), and \( z \).
The variables in an equation are typically written in an order that is easily interpreted by the reader. While it would not be incorrect in Example 1 to say that the variables were \( y \), \( z \), and \( x \) with coefficients \(1 \), \( -8\), and \(2\), it would be unnecessarily confusing to do so.
We often number the variables using subscripts, especially when we have a large number of variables to deal with.
Example 2. Consider the equation \(-x_1 + \pi x_3 - \sqrt{2} x_4 = 0.07 \). The coefficients are \( -1\), \( 0\), \(\pi\), and \(-\sqrt{2}\), with variables \( x_1, \ldots, x_4 \).
The previous example illustrates a couple of points that are important to keep in mind when considering linear equations:
Example 3. Consider the equation \( 2x-y+6 = 13-4x \). We can rewrite this equation as \( 6x-y=7\) to see that it is linear with coefficients \(6\) and \(-1\) with variables \(x\) and \(y\).
Note that we could just as easily have written the equation as \( -6x+y = -7\), which changes the coefficients we see in the equation. This shows us that linear equations can be written in many equivalent forms. However, as we will see, writing an equation in a different form does not change what solution(s) it does or does not have.
We also see from Example 3 that a linear equation does not have to be written in the form shown in equation (1). Rather, it must be able to be written in that form.
Example 4. The equation \( \sqrt{x} + \frac{1}{y} = wz - 10 \) is not linear. It cannot be written in the form specified in equation (1), no matter what algebraic operations we do to both sides.
Definition. A solution of an equation (or "to" an equation) is a value for each variable that makes the equation true.
Example 5. Consider the equation \( x-4y-6z = 17\). One solution of this equation is \(x=3\), \(y=4\), and \(z=-5\), since \( (3)-4(4)-6(-5) = 17 \).
Most linear equations have infinitely many solutions. One of the topics we will discuss in Lecture 10 is how to describe solutions of a single linear equation in a useful way. However, it is a more interesting problem to consider systems of linear equations and ask what values of the variables are simultaneously solutions of each equation.
Definition. A system of linear equations (also called a "linear system") is a collection of one or more linear equations involving the same variables.
Example 6. Here is an example of a system of two equations involving the variables \( x_1, x_2\), and \(x_3\): \begin{alignat*}{5} 2x_1 && -x_2 && +1.5x_3 && = && 8 \\ x_1 && &&-4x_3 && = && -7 \end{alignat*}
Definition. A solution of a system of linear equations is a value for each variables that makes each equation in the system true simultaneously. Solutions are often (but not always) written as ordered tuples \( (s_1, s_2, \ldots, s_n) \), where \( s_1 \) is the value of \( x_1 \), \(s_2\) is the value of \(x_2\), etc.
Example 6, continued. We see that \( (-3, -8, 4) \) is not a solution to the system in Example 6. The first equation is true, since \( 2(-3) -(-8) + 1.5(4) = 8 \). However, the second equation is not true, since \( (-3)-4(4) \neq -7 \). Since \( (-3, -8, 4) \) does not make every equation in the system true, it is not a solution of the system.
By contrast, the ordered triple \( (5, 6.5, 3) \) is a solution to the system in Example 6, since it makes each of the equations in the system true simultaneously: \begin{alignat*}{5} 2(5) && -(6.5) && +1.5(3) && = && 8 \\ (5) && &&-4(3) && = && -7 \end{alignat*}
At this point, there are several natural questions to ask about systems of linear equations:
We will consider these questions and answer them in the coming lectures.
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