1.1: Solutions and Elementary Operations (2024)

Practical problems in many fields of study—such as biology, business, chemistry, computer science, economics, electronics, engineering, physics and the social sciences—can often be reduced to solving a system of linear equations. Linear algebra arose from attempts to find systematic methods for solving these systems, so it is natural to begin this book by studying linear equations.

If \(a\), \(b\), and \(c\) are real numbers, the graph of an equation of the form

\[ax + by = c \nonumber \]

is a straight line (if \(a\) and \(b\) are not both zero), so such an equation is called a linear equation in the variables \(x\) and \(y\). However, it is often convenient to write the variables as \(x_1, x_2, \dots, x_n\), particularly when more than two variables are involved. An equation of the form

\[a_1x_1 + a_2x_2 + \dots + a_nx_n = b \nonumber \]

is called a linear equation in the \(n\) variables \(x_1, x_2, \dots, x_n\). Here \(a_1, a_2, \dots, a_n\) denote real numbers (called the coefficients of \(x_1, x_2, \dots, x_n\), respectively) and \(b\) is also a number (called the constant term of the equation). A finite collection of linear equations in the variables \(x_1, x_2, \dots, x_n\) is called a system of linear equations in these variables. Hence,

\[2x_1 - 3x_2 + 5x_3 = 7 \nonumber \]

is a linear equation; the coefficients of \(x_1\), \(x_2\), and \(x_3\) are \(2\), \(-3\), and \(5\), and the constant term is \(7\). Note that each variable in a linear equation occurs to the first power only.

Given a linear equation \(a_1x_1 + a_2x_2 + \dots + a_nx_n = b\), a sequence \(s_1, s_2, \dots, s_n\) of \(n\) numbers is called a solution to the equation if

\[a_1s_1 + a_2s_2 + \dots + a_ns_n = b \nonumber \]

that is, if the equation is satisfied when the substitutions \(x_1 = s_1, x_2 = s_2, \dots, x_n = s_n\) are made. A sequence of numbers is called a solution to a system of equations if it is a solution to every equation in the system.

For example, \(x = -2\), \(y = 5\), \(z = 0\) and \(x = 0\), \(y = 4\), \(z = -1\) are both solutions to the system

\[ \begin{array}{rrrrrrr} x & + & y & + & z & = & 3\\ 2x & + & y & + & 3z & = & 1 \end{array} \nonumber \]

A system may have no solution at all, or it may have a unique solution, or it may have an infinite family of solutions. For instance, the system \(x + y = 2\), \(x + y = 3\) has no solution because the sum of two numbers cannot be 2 and 3 simultaneously. A system that has no solution is called inconsistent; a system with at least one solution is called consistent. The system in the following example has infinitely many solutions.

The quantities \(s\) and \(t\) in Example\(\PageIndex{1}\) are called parameters, and the set of solutions, described in this way, is said to be given in parametric form and is called the general solution to the system. It turns out that the solutions to every system of equations (if there are solutions) can be given in parametric form (that is, the variables \(x_1\), \(x_2\), \(\dots\) are given in terms of new independent variables \(s\), \(t\), etc.). The following example shows how this happens in the simplest systems where only one equation is present.

When only two variables are involved, the solutions to systems of linear equations can be described geometrically because the graph of a linear equation \(ax + by = c\) is a straight line if \(a\) and \(b\) are not both zero. Moreover, a point \(P(s, t)\) with coordinates \(s\) and \(t\) lies on the line if and only if \(as + bt = c\)—that is when \(x = s\), \(y = t\) is a solution to the equation. Hence the solutions to a system of linear equations correspond to the points \(P(s, t)\) that lie on all the lines in question.

In particular, if the system consists of just one equation, there must be infinitely many solutions because there are infinitely many points on a line. If the system has two equations, there are three possibilities for the corresponding straight lines:

1. The lines intersect at a single point. Then the system has a unique solution corresponding to that point.
2. The lines are parallel (and distinct) and so do not intersect. Then the system has no solution.
3. The lines are identical. Then the system has infinitely many solutions—one for each point on the (common) line.

See Also

9.1.1: How Solutions Form

These three situations are illustrated in Figure \(\PageIndex{1}\). In each case the graphs of two specific lines are plotted and the corresponding equations are indicated. In the last case, the equations are \(3x - y = 4\) and \(-6x + 2y = -8\), which have identical graphs.

With three variables, the graph of an equation \(ax + by + cz = d\) can be shown to be a plane (see Section [sec:4_2]) and so again provides a “picture” of the set of solutions. However, this graphical method has its limitations: When more than three variables are involved, no physical image of the graphs (called hyperplanes) is possible. It is necessary to turn to a more “algebraic” method of solution.

Before describing the method, we introduce a concept that simplifies the computations involved. Consider the following system

\[ \begin{array}{rlrlrlrcr} 3x_1 & + & 2x_2 & - & x_3 & + & x_4 & = & -1\\ 2x_1 & & & - & x_3 & + & 2x_4 & = & 0\\ 3x_1 & + & x_2 & + & 2x_3 & + & 5x_4 & = & 2 \end{array} \nonumber \]

of three equations in four variables. The array of numbers¹

\[\left[ \begin{array}{rrrr|r} 3 & 2 & -1 & 1 & -1 \\ 2 & 0 & -1 & 2 & 0 \\ 3 & 1 & 2 & 5 & 2 \end{array} \right] \nonumber \]

occurring in the system is called the augmented matrix of the system. Each row of the matrix consists of the coefficients of the variables (in order) from the corresponding equation, together with the constant term. For clarity, the constants are separated by a vertical line. The augmented matrix is just a different way of describing the system of equations. The array of coefficients of the variables

\[\left[ \begin{array}{rrrr} 3 & 2 & -1 & 1 \\ 2 & 0 & -1 & 2 \\ 3 & 1 & 2 & 5 \end{array} \right] \nonumber \]

is called the coefficient matrix of the system and \(\left[ \begin{array}{r} -1 \\ 0 \\ 2 \end{array} \right]\) is called the constant matrix of the system.