Systems of Linear Equations

Section 1.1 Systems of Linear Equations

We will introduce the first set of big ideas in this book with an example.

Example 1.1.1.

Alex reaches into his pocket and pulls out a handful of coins. He tells us that he’s holding 90 cents in his hand, consisting of only nickels (worth five cents each) and quarters (worth 25 cents each). How many coins of each type is Alex holding?

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We can model this information with a linear equation. Let $x$ be the number of nickels in Alex’s hand, and let $y$ be the number of quarters. The equation that captures the information Alex shared is

\begin{equation} 5x + 25y = 90\text{.}\tag{1.1} \end{equation}

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In this case, we don’t have enough information to answer Alex’s question. It could be that Alex is holding three nickels and three quarters ($15+75=90$) or that he is holding 13 nickels and one quarter ($65+25=90$). There are quite a few solutions to equation (1.1). (Note that in this example, since it would not make sense to have only part of a coin, we need our values of $x$ and $y$ to be non-negative integers.)

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Now imagine that Alex gives us additional information by telling us that he is holding exactly ten coins. We can put this information into a second linear equation, and we now have what is called a system of equations. We need to know what values of $x$ and $y$ satisfy the following equations simultaneously:

\begin{align*} 5x+25y \amp= 90 \\ x+y \amp= 10 \text{.} \end{align*}

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A small amount of work shows us that the values $x=8$ and $y=2$ satisfy these equations simultaneously. (The reader should not worry right now about where those numbers came from. We’ll get there soon.) This means that Alex is holding eight nickels and two quarters.

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The first major task of Linear Algebra is to learn how to handle systems of linear equations like the one given in Example 1.1.1. We will learn a method to analyze such systems and determine their solutions (if solutions exist). We need a number of definitions as we get started.

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Note 1.1.2.

We will use the symbol $\mathbb{R}$ to denote the set of all real numbers. At the beginning of our work, we will be using $\mathbb{R}$ as our set of numbers for almost everything. However, when we get to Section 2.1, we will move away from $\mathbb{R}$ to a more general description of a set of numbers that “works” for solving linear equations.

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Definition 1.1.3.

A linear equation in the variables $x_1, \ldots, x_n$ is one which can be written in the form

\begin{equation*} a_1x_1 + \cdots + a_nx_n = b, \end{equation*}

where the $a_i$ and $b$ are constants. The numbers $a_1,\ldots,a_n,b$ all come from $\mathbb{R}\text{.}$ In the special situation where $b=0\text{,}$ this is called a homogeneous linear equation. (Note that when there are only two variables in view we may use $x$ and $y$ instead of $x_1$ and $x_2\text{;}$ similarly, when there are only three variables present, we may use $x\text{,}$ $y\text{,}$ and $z\text{.}$)

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The word “linear” in the phrase “linear equation” should make us think of a single power of a variable. In a linear equation, therefore, we will have no terms involving $x_1^4\text{,}$ $\sin(x_2)\text{,}$ $x_2x_3\text{,}$ $\sqrt{x_4}\text{,}$ or anything other than a single power of a variable.

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Example 1.1.4.

The following are linear equations:

\begin{equation*} 4x + 2y = 98, \hspace{.3in} -2x_1 -6x_2 + 5x_3=0, \hspace{.3in} 14x_1 - 15x_3 = -7\text{.} \end{equation*}

These are all linear equations because every appearance of a variable term in these equations contains only a single variable raised to the first power.

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The following are not linear equations:

\begin{equation*} 4x^2-y=9, \hspace{.3in} \ln(x_1) + 3x_2 = 2, \hspace{.3in} x_1x_2 - \tan(x_3) = 0\text{.} \end{equation*}

These are not linear equations because each equation has at least one variable term with something other than a single variable to a single power.

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Definition 1.1.5.

A system of linear equations is a collection of one or more equations involving the same variables. (We sometimes shorten this and refer to a linear system.) When all of these linear equations are homogeneous linear equations, this may be called a homogeneous linear system.

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Example 1.1.6.

Here is a system of linear equations involving the variables $x_1\text{,}$ $x_2\text{,}$ $x_3\text{,}$ and $x_4\text{:}$

\begin{align*} 2x_1 - x_3+8x_4 \amp = 10\\ -x_1 + 3x_2 - 6x_4 \amp = -4\text{.} \end{align*}

Note that not all variables need to be present in each equation. When a variable is missing, we consider that variable to have a zero coefficient in that equation. It may be convenient (and preferred) to align the terms with the same variables vertically, but when an equation lacks one of the variables this creates a blank horizontal space.

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Definition 1.1.7.

A solution to a system of linear equations in $n$ variables is a list of $n$ numbers, $(c_1,\ldots,c_n)\text{,}$ such that when the corresponding variables are assigned these numeric values (plug $c_1$ in everywhere for $x_1\text{,}$ plug $c_2$ in everywhere for $x_2\text{,}$ and so on), all the equations are true statements. The set of all solutions is called the solution set of the linear system. Two linear systems are said to be equivalent if their solution sets are equal.

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The language of solution sets and equivalent linear systems may seem unnecessarily complex. However, the method (see Algorithm 1.3.8) for solving a linear system is much easier to describe with these terms firmly in hand.

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Example 1.1.8.

Consider the following linear system:

\begin{align*} 2x+y \amp=-1 \\ x-3y \amp=17 \text{.} \end{align*}

A solution to this system is $(2,-5)\text{.}$ (We use this notation to mean $x=2$ and $y=-5\text{.}$) We verify this claim by plugging these numbers in for the variables and checking that both equations turn out to be true. (In fact, this is the only solution to this system.)

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The fact that we can write down a system of linear equations does not mean that it has a solution. Many linear systems have no solution at all. Others have one or an infinite number of solutions. This can be illustrated by some examples of lines graphed in the familiar setting of $\mathbb{R}^2\text{.}$

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Example 1.1.9.

We first consider the linear system consisting of two simple equations where the solution to each equation is a line in $\mathbb{R}^2\text{:}$

\begin{align*} 2x+y \amp=5 \\ -x+y \amp=-1 \text{.} \end{align*}

Readers will likely realize that the solution to this system happens when the graphs of these lines intersect. We see from the graph below that the intersection occurs at the point $(2,1)\text{.}$ Since this is the only intersection point, there is only one solution to this system.

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The graph of two lines on coordinate axes. The lines have slopes 1 and -2 and meet at the point (2,1). See long description. — Figure 1.1.10. A graph of two intersecting lines
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We now consider a second linear system:

\begin{align*} 2x+y \amp=5 \\ 2x+y \amp=3 \text{.} \end{align*}

The graphs of these two lines appear below, but in this case the lines do not intersect at all because they are parallel. (The slope of both lines is $-2\text{.}$)

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The graph of two lines on coordinate axes. The lines both have slope -2 and are parallel. See long description. — Figure 1.1.11. A graph of two parallel lines
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Here is a third linear system in two variables:

\begin{align*} 2x+y \amp=5 \\ 6x+3y \amp=15 \text{.} \end{align*}

Each of these equations has a solution set whose graph is a line. In this particular case, we obtain the same line for both equations, which means that the linear system has an infinite number of solutions. Each one of the infinite solutions to the first equation is a solution to the second equation, and vice versa.

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The graph of one line on coordinate axes. The line has slope -2 and y intercept 5. See long description. — Figure 1.1.12. A graph of two indentical lines
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What we saw in Example 1.1.9 is no accident. This simple collection of examples in $\mathbb{R}^2$ exposes our need for the following definitions.

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Definition 1.1.13.

A system of linear equations is consistent if it has at least one solution. A system is inconsistent if it has no solutions. When a linear system has only one solution we say that the solution is unique.

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This definition and the principle illustrated in Example 1.1.9 mean that whenever we encounter a system of linear equations we have two questions to ask. Is this system consistent? If the system is consistent, is the solution unique?

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Reading Questions Reading Questions

1.

This question is designed to help you understand linear equations.

Write down an example of a linear equation involving the variables $x\text{,}$ $y\text{,}$ and $z\text{.}$
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Write down an example of an equation involving the variables $x\text{,}$ $y\text{,}$ and $z$ which is not linear.
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2.

Find the solution set to the following linear system:

\begin{align*} 3x + 7y \amp = 9 \\ -2x + 6y \amp = -2\text{.} \end{align*}

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Exercises Exercises

1.

Carlos has ten coins totalling $\$1.10\text{.}$ Each coin is either a nickel, a dime, or a quarter. He has twice as many nickels as he has quarters. How many of each coin does Carlos have?

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Model this problem with a linear system. (You do not need to find the solution set of this linear system.)

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2.

Suppose that $f(x) = ax^2+bx+c$ is a function whose graph passes through the points $(-1,6)\text{,}$ $(1,0)\text{,}$ and $(3,-2)\text{.}$ What are the values of $a\text{,}$ $b\text{,}$ and $c\text{?}$

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Model this problem with a linear system. (You do not need to find the solution set of this linear system.)

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Answer.

The linear system to solve is

\begin{align*} a - b + c \amp =6\\ a+b+c \amp =0\\ 9a+3b+c \amp =-2\text{.} \end{align*}

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3.

For each part, determine whether the equations form a linear system.

\begin{align*} 3y - 4z + x \amp = \sqrt{2}\\ -z + 2w + 7x \amp = 14 \end{align*}

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\begin{align*} 2x - 10y + \frac{3}{z} \amp = 16\\ 7x + 12z \amp = -1 \end{align*}

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\begin{align*} x - e^4 z \amp = 9\\ \ln(8)x + 13y \amp = -3 \end{align*}

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4.

Determine whether or not each given list is a solution to this linear system.

\begin{align*} x_1 - 2x_2 + 8x_3 \amp = -5\\ -x_1 + 3x_2 - 10x_3 \amp = 6\\ 2x_1 - 3x_2 + 14x_3 \amp = -9\text{.} \end{align*}

$\displaystyle (-7, 3, 1)$
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$\displaystyle (-5,0,0)$
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$\displaystyle (-4,4,1)$
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$\displaystyle (5,-3,-2)$
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5.

Determine whether or not each given list is a solution to this linear system.

\begin{align*} x - y + 3z \amp = -6\\ -3x + 4y - 10z \amp = 22\\ -2x + 4y -8z \amp = 21\text{.} \end{align*}

$\displaystyle (1, 1, -2)$
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$\displaystyle (-2,3,4)$
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$\displaystyle (0,3,-1)$
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Answer.

No.
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No.
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No.
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6.

For the following linear systems, graph the solution set of each equation. Then graph the solution set of the linear system.

\begin{align*} -2x+y \amp =-3\\ x-5y \amp = 7 \end{align*}

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\begin{align*} 2x-2y \amp = 5\\ 3x+y \amp = -2\\ -x-y \amp = 1 \end{align*}

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\begin{align*} 2x-3y \amp = -2\\ -2x+3y \amp = -3 \end{align*}

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\begin{align*} 2x - 4y \amp = -1\\ -x+2y \amp = \tfrac{1}{2} \end{align*}

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\begin{align*} x+4y \amp = 5\\ 3x-y \amp = 0\\ -2x-8y \amp = -10 \end{align*}

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Answer.

The solution set is $\{(8/9,-11/9)\}\text{.}$
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The solution set is empty.
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The solution set is empty.
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The solution set is $\{(x,y) \mid y = \frac{1}{2}x + \frac{1}{4}\}\text{.}$
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The solution set is $\{(5/13,15/13)\}\text{.}$
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Writing Exercises

7.

For each of the following, write an example of a linear system with of two equations and two variables with the given property. (The system must be one you’ve not yet seen!) Explain why your example has the property.

The system has no solutions.
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The system has exactly one solution.
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The system has infinitely many solutions.
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Answer.

This system has no solution because the lines are parallel:

\begin{align*} -2x + y \amp =0\\ -2x + y \amp =1\text{.} \end{align*}

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This system has one solution because the lines only intersect at the origin:

\begin{align*} -2x + y \amp =0\\ 2x+y \amp =0\text{.} \end{align*}

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This system has infinitely many solutions because the equations describe the same line:

\begin{align*} -2x+y \amp =0\\ 2x-y \amp =0\text{.} \end{align*}

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8.

Consider a linear system of $m$ equations and $n$ variables and suppose that $(c_1,\ldots,c_n)$ and $(d_1,\ldots,d_n)$ are both solutions to this system. Under what conditions is $(c_1+d_1, \ldots, c_n+d_n)$ also a solution to the same system? Explain.

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Answer.

This is true if and only if the system is a homogeneous linear system.

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9.

Consider a linear system of $m$ equations and $n$ variables and suppose that $(c_1,\ldots,c_n)$ and $(d_1,\ldots,d_n)$ are both solutions to this system. Let $t$ be a real number. Prove that $(tc_1 + (1-t)d_1, \ldots, tc_n + (1-t)d_n)$ is also a solution to this same linear system.

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