Linear Equations & their Solutions

Transcription

1 CHAPTER Linear Equations & their Solutions. Blah Blah Linear equations arise everywhere. They provide neat ways of expressing simple relationships between certain quantities, or variables. Virtually every Linear Algebra question which we will come across can be paraphrased as a question about linear equations, so it is important to get comfortable with the language. This chapter introduces an algorithmic method, called row reduction, for solving any system of linear equations. The method, although sometimes messy in calculations, is very simple in its instructions. By the end of the chapter, you will know how to solve any system of linear equations! And it turns out that row reduction is useful in other matrix calculations later on. The basic idea behind it is that we can perform certain operations to a linear equation and have that equation still carry the same information. These operations entail multiplying both sides of an equation by a number, and adding one equation to another. For example, x = 4and 3x = 6bothsaythe same thing, that x =. The information gets more complicated of course, but the idea remains the same: we can simplify equations and still maintain the same solutions (x =4 x=).. Preliminaries If you re reading this, you probably already have some sense of what linear means. For instance, the two equations x + y + z = 3 3x + y z = are linear. We can even consider them together as a system of linear equations. An equation can be shuffled to some degree and still maintain its linearity. Both y =3 x zand x + y + z 3= are linear equations; however, (x + y + z) = 9 is not. Here are some more examples: Linear Not linear 3 z =3 x y = x+y+z ax + by = c ax + y = c x y + z =3x+y x+y=xy 5x +3x+= 5x +3x+= sin(x) 3cos(y)= sin(x) 3cos(y)= The last two lines are not typos! While 5x +3x+ = is indeed a quadratic equation in terms of x, we can also think of it as the equation 5A +3B+ = where A = x and B = x. Thatis, the equation is linear in x and x. Even though x depends on x, the expression itself is linear: it consists of numbers multiplied by objects and then added all up. Similarly, sin(x) 3cos(y) = is linear in sin(x) andcos(y), but not in x and y. This might feel like splitting hairs, but it is an important consideration which we will make use of in later chapters. For now, let us try to formalize this discussion. DEFINITION. A linear equation in n unknowns is an equation which can be written as a x + a x + +a n x n =b The x i s are called unknowns (or variables), and the a i s (coefficients) andb(constant term) are numbers.

2 . LINEAR EQUATIONS & THEIR SOLUTIONS The notation gets a bit messy when looking at a system of m equations in n unknowns: a x + a x + + a n x n = b a x + a x + + a n x n = b.. a m x + a m x + + a mn x n = b m By convention, the row subscript is written first, the column subscript second. The constant terms usually go on the right hand side of the equations. EXAMPLE. In the following system of 4 equations in 3 unknowns, x y = 3x + y + z = 7x + 3y + 5z = 4 x + y z = a 3 =,a 3 =7,a 43 =, a =,andb 3 = 4; while a 34 does not exist. The unknowns are x, y, and z. Note that the unknowns with zero coefficients are omitted. Whenever you see an equation, you want to solve it; that has so far been your mathematical conditioning! This book will not offer you a way out of the cycle. In fact, it will only help reinforce it. Our primary interest when dealing with linear equations is the set of solutions. Consider the following system of two equations in two unknowns: x y = 5 x + 3y = 3 We could solve for x in the first equation {x =5+y}and substitute that back into the second equation {(5 + y) + 3y = 3}. Now we can easily solve for y { + 4y + 3y = 3 = 7y= 7 = y = } and then back-substitute in for x {x =5+( ) = 3}. Wehavefoundthe solution: x =3,y =. This method would probably work for any system of linear equations in unknowns. But as the number of equations and unknowns increase, the need for a cleaner method becomes more apparent. Before we develop that method, we make another formal definition. DEFINITION 3. A solution to a system of m equations in n unknowns (call the unknowns x, x,, x n )isasetofnnumbers x = c, x = c,, x n = c n which satisfy all m equations at the same time. A system of equations is consistent if it has at least one solution; otherwise, it is inconsistent. One can check that example has a solution x =,y=,z= ; in fact, that is the only solution to example. On the other hand, example 6 below is inconsistent since the last equation cannot possibly be satisfied. It is also possible for a system of equations to have more than one solution. In fact, if there is more than one solution, then there are infinitely many solutions. EXAMPLE 4. The following system has infinitely many solutions. x y + z = 3 x + y + z = Can you find them all? 3. Row Reduction Let us revisit the system from moments ago, and solve it in a seemingly more involved manner. x y = 5 x + 3y = 3

3 3. ROW REDUCTION 3 Note that if an equation is multiplied by a constant, or if two equations are added together, then the equalities remain. In other words, we do not change the solutions to a system of linear equations if we do the same thing to both sides of the equation. For instance, let us add timesthefirst equation to the second equation. This will eliminate the x from the second equation. x y = 5 7y = 7 Note that we left the first equation intact. Now, we can multiply the second equation by 7. x y = 5 y = And finally, we can add times the second equation to the first. x = 3 y = You just saw a simple example of row reduction. Fuzzily speaking, i.e. skipping definitions, row reduction (RR) will work as follows: System of equations RR Echelon Form RR Reduced echelon Form As you can see, some definitions are in order. DEFINITION 5. Asystemofmequations in n unknowns is in echelon form if the first nonzero entry in each row is to the right of the first non-zero coefficient in the row above it. The pivots in a system of linear equations in echelon form are the first non-zero coefficients of each row (equation). As is often the case, the definition of echelon form is much easier to grasp through an example. Example is not in echelon form, since the first non-zero entry in the second row is 3x, andthatis not to the right of the first non-zero entry in the row above it, x. EXAMPLE 6. The following is a system in echelon form. x x + 3x 3 + x 4 = x x 3 x 4 = 3 5x 4 = = The pivots here are,, and 5. Recall that this system is inconsistent, since there is no solution which can satisfy the last equation. If you stare at the definition for a short while, you will see that if a system is in echelon form, then all the entries below a pivot must equal zero. Sometimes people insist that in echelon form, all the pivots must equal ; we will not be so strict. Echelon form is a useful halfway stop on the way to finding the solutions to a system of linear equations, since we can always tell whether a system is consistent or not once we transform it to that form. There are three basic steps which, in combination, will allow us to transform any given system of equations thusly. DEFINITION 7. A row operation on a system of linear equations is one of the following: adding a multiple of one row to another row; switching two rows; multiplying one row by a non-zero number. None of these operations changes what the solutions are. For instance, x =,y= 3,z= 4 is a solution to example 4 as well as to 3 x y + z = 3 x y + 4z = 4

4 4. LINEAR EQUATIONS & THEIR SOLUTIONS Can you tell which row operation was performed on example 4 to obtain the above system? The purpose of row operations is to transform a system of linear equations into one in echelon form without affecting the solutions (it turns out that finding solutions becomes easier if a system is in echelon form). DEFINITION 8. Row reduction is a process which repeatedly uses row operations to transform a system of linear equations into echelon form. Row reduction is also called Gaussian elimination. If you are interested in programming, you might want to try coding the row reduction process as efficiently as possible. There are many such programs and applets available on-line. Nevertheless, coding the algorithm will give you insight into its mechanics. We have yet to define and discuss reduced echelon form. It might be helpful to first take a look at row reduction in action. EXAMPLE 9. Consider the following system of 4 equations in 5 unknowns: x + x + 3x 4 x 5 = 3x + 6x + x 4 + 4x 5 = 7 x + x x 3 + 5x 4 3x 5 = 3 x + 4x + x 3 x 4 + 3x 5 = If we are to reduce it to echelon form, we must consider the first non-zero entry in the first row, x. The first pivot is therefore the top left coefficient, namely. We recall that all the entries below a pivot must equal zero; we proceed with that first goal in mind. Add 3 times the first row to the second row: x + x + 3x 4 x 5 = 3x 4 + 7x 5 = x + x x 3 + 5x 4 3x 5 = 3 x + 4x + x 3 x 4 + 3x 5 = Observe that the first row was left intact. Now add times the first row to the third row: x + x + 3x 4 x 5 = 3x 4 + 7x 5 = x 3 + x 4 x 5 = x + 4x + x 3 x 4 + 3x 5 = There is one more row reduction step left to acheive the first goal. Add times the first row to the fourth row: x + x + 3x 4 x 5 = 3x 4 + 7x 5 = x 3 + x 4 x 5 = x 3 7x 4 + 5x 5 = 4 All entries below the first pivot now equal zero. The second row poses a problem: if 3 is to be the second pivot, then there is no way of eliminating the entries x 3 and x 4 inrows3and4viarow reduction. This can be solved by switching rows and 3: x + x + 3x 4 x 5 = x 3 + x 4 x 5 = 3x 4 + 7x 5 = x 3 7x 4 + 5x 5 = 4 We could have just as easily switched rows and 4 towards the same end. Now the second pivot can be. The corresponding entry below in row three the coefficient of x 3 is already zero. Add

5 4. REDUCED ECHELON FORM 5 times the second row to the fourth row: x + x + 3x 4 x 5 = x 3 + x 4 x 5 = 3x 4 + 7x 5 = 3x 4 + x 5 = The third pivot is now 3. Add the third row to the fourth row: x + x + 3x 4 x 5 = x 3 + x 4 x 5 = 3x 4 + 7x 5 = 8x 5 = The system has been transformed into echelon form by row reduction. The pivots are,, 3 and 8. From the echelon form, we can figure out whether a system of linear equations is consistent or not. The only way a system of linear equations can be inconsistent is if its echelon form has a least one row with all zero coefficients (that is, without pivots) but with a non-zero constant term; see for instance example 6. We will look at more such examples later on. Once we determine that a system of equations is consistent, we can further reduce it to its bare essentials. A system of equations in reduced echelon form will divulge all its secrets. 4. Reduced Echelon Form DEFINITION. A system of linear equations is in reduced echelon form if, in addition to being in echelon form, all pivots equal, and all entries above (as well as below) the pivots equal zero. So a system is in reduced echelon form when the same process is applied to the entries above the pivots that was applied to the entries below them. EXAMPLE 9 (continued) Further row reduction can transform the above system into reduced echelon form. All the pivots can be changed to s by multiplying the second row by, the third row by 3,andthefourthrowby 8 : x + x + 3x 4 x 5 = x 3 x 4 + x 5 = x x 5 = 3 x 5 = 8 We can now eliminate all the entries above the pivot in the fourth row. Add 7 3 times the fourth row to the third row, times the fourth row to the second row, and times the fourth row to the first row: x + x + 3x 4 = 5 8 x 3 x 4 = 3 4 x 4 = x 5 = 8 Finally, add times the third row to the second row, and 3 times the third row to the first row: x + x = x 3 = 5 x 4 = 8 x 5 = 8 Since the entry above x 3 was already zero, our work is done. This system is in reduced echelon form! It is easy to see the solution includes x 5 = 8, x 4 = 5 8,andx 3=. So the reduced echelon form is already paying off. But what about x and x? We know that x = x, but what does x equal? That question was rhetorical: x can equal whatever it wants, and x is bound by that 5 8

6 6. LINEAR EQUATIONS & THEIR SOLUTIONS result. If x =thenx =,ifx = 3thenx =6. Ifx =πthen x = π. In fact, this system of equations has infinitely many solutions, as predicated by the freedom of x. 5. General Solutions We are almost done with laying the groundwork, with developing all the requisite language of linear equations. DEFINITION. After row reduction of a system of linear equations to echelon form, if a variable corresponds to a column with a pivot, it is called basic. Otherwise, it is called free. The rank of a system of linear equations is the number of basic variables it has (or equivalently, the number of pivots). OBSERVATION. The rank of a system of m equations in n unknowns cannot be greater than m or n, and hence, cannot be greater than min(m, n). This is because each row of the system can have at most one pivot (hence rank m) and each column can also have at most one pivot (hence rank n). EXAMPLE 9 (continued) x is a free variable. All the other variables (x, x 3, x 4,andx 5 )are basic. The rank is therefore 4. Recall that a solution to a system of m equations in n unknowns is a set of n numbers x = c, x = c,, x n =c n which satisfy all m equations. Also recall that row reduction does not change the solutions to systems of linear equations. That is, any solution to a reduced echelon form is also a solution to the original system, and vice versa. DEFINITION 3. The general solution to a system of linear equations is the set of all possible solutions to that system. It can be calculated by first determining the reduced echelon form of the system, and then writing all the basic variables in terms of the free variables. EXAMPLE 9 (continued) By looking at the reduced echelon form of the system, it is easy to see that the general solution is: x = x x = x x 3 = 5 x 4 = 8 x 5 = 8 For each value of x there is a different, unique solution. The line x = x, though redundant, simply asserts that x is free. It might be more intuitive to write it a bit differently: x = t x = t x 3 = x 4 = 5 8 x 5 = 8 EXAMPLE 4. In order to find the general solution to the following system of 4 equations in 6 unknowns, we first find its echelon form. The row reduction descriptions have been omitted for

7 5. GENERAL SOLUTIONS 7 your guessing pleasure. x + x x 3 + x 4 + x 5 4x 6 = x 3x + x 3 x 4 4x 5 + 3x 6 = 7 x + x + x 4 + 3x 5 9x 6 = 5 3x + x 5x 3 + 3x 4 + x 5 8x 6 = x + x x 3 + x 4 + x 5 4x 6 = x x 3 x 5 + 5x 6 = 3 x + x 3 + x 5 5x 6 = 3 x x 3 x 5 + 4x 6 = 6 x + x x 3 + x 4 + x 5 4x 6 = x x 3 x 5 + 5x 6 = 3 = x 5 6x 6 = x + x x 3 + x 4 + x 5 4x 6 = x x 3 x 5 + 5x 6 = 3 x 5 6x 6 = = The system is now in echelon form. We can already see that the basic variables are x, x and x 5, whereas x 3, x 4 and x 6 are free. The rank equals 3. The system is consistent since the only row without pivots is of the form =. We will keep that row while we reduce further, as a reminder that the system started out with 4 equations. On to reduced echelon form: x + x x 3 + x 4 + x 5 4x 6 = x + x 3 + x 5 5x 6 = 3 x 5 3x 6 = = x + x x 3 + x 4 x 6 = x + x 3 + x 6 = 3 x 5 3x 6 = = x x 3 + x 4 x 6 = x + x 3 + x 6 = 3 x 5 3x 6 = = We are now ready to read off the general solution to the system, by writing the basic variables in terms of the free ones. x = + t u + v x = 3 t v x 3 = t x 4 = u x 5 = 3v x 6 = v Even though the systems of linear equations in examples 9 and 4 both have infinitely many solutions, the second one has, in a sense, more solutions that the first. We will explore these

8 8. LINEAR EQUATIONS & THEIR SOLUTIONS different levels of infinity once we discuss dimension. For the time being, let us simply note that the free variables provide a measure of how large the set of solutions is. 6. New Notation & More Examples It is not necessary to keep track of the variable names when performing row reduction. In fact, it is more efficient to ignore them completely, since only the coefficients affect the calculations. For instance, the row reduction of example 4 can be more succinctly denoted by: etc EXAMPLE 5. We tackle the following system of equations using the new notation. u + v = u + 3v = 3 u v = 3 From the third line ( = 3) we conclude that the system is inconsistent, that is, there are no solutions. We can therefore stop the row reduction process at echelon form. EXAMPLE 6. Yet another system, this time of 3 equations in 3 unknowns. x y + 3z = 3 3x + z = x + y = The system is now in echelon form, and so we can already draw some conclusions. Note that the system is consistent; compare with the inconsistency of example 5. Also note that all three variables are basic, so we can expect there to be only one solution (why?). 3 3 So the only solution is x = y = z = 3 3 6

9 8. THE GEOMETRY OF LINEAR EQUATIONS 9 7. Homogeneous Equations It is usually impossible to tell whether a given system of linear equations is consistent or not without going through the row reduction process first, at least all the way to echelon form. However... DEFINITION 7. A system of linear equations is called homogeneous if all the constant terms equal zero: a x + a x + + a n x n = a x + a x + + a n x n =.. a m x + a m x + + a mn x n = OBSERVATION 8. A homogeneous system of equations is always consistent. Namely, x =,x =,,x n = is always a solution. It can have other solutions as well, but the presence of this trivial solution assures consistency. EXAMPLE 9. Consider the following homogeneous system of 3 equations in 3 unknowns. s u = s + t + u = 3s + t u = 3 Therankis,andthefreevariableisu. So the general solution is s = a/ t = a/ u = a Note that if u =,thens=andt=. This is the specific, trivial solution that we always expect to be part of the general solution of a homogenous system. 8. The Geometry of Linear Equations If you have studied analytic geometry before and who hasn t? you will recall that an equation in unknowns defines a line in R,orthexy-plane. For instance, 6x +y= is a straight line with slope 3 andy-intercept. This line is the geometric representation of the general solution to this system of equation in unknowns, in other words, each point (x,y )on the line satisfies the linear equation. Solving a system of m equations in unknowns is the same as finding all points (x, y) inr where all m straight lines intersect simultaneously. If the system is inconsistent, then there is no point through which all the lines pass. If the system is consistent, then there is either one solution (all the lines have exactly one point in common, though some of the lines might coincide) or there are infinitely many solutions (all m equations actually define the same line). Figure illustrates these cases for three lines in R.

10 . LINEAR EQUATIONS & THEIR SOLUTIONS Figure (a): Inconsistent 3 lines coinciding lines coinciding One solution Infinitely many solutions Figure (b): Consistent Analogously, an equation in 3 unknowns defines a plane in R 3. This might not be as familiar an object to you as lines are, but the geometry could still make sense on an intuitive level. For instance, x + y z =4 is a plane with normal vector (,, ) and z-intercept 4 (the normal vector is the analogue of the slope). So solving a system of m equations in 3 unknowns is the same as finding all points (x, y, z) inr 3 where all m planes intersect simultaneously. If the system is inconsistent, then there is no point through which all the planes pass. If the system is consistent, then there is either one solution (all the planes have exactly one point in common) or there are infinitely many solutions (either all the planes coincide, or they intersect in a straight line). Figure illustrates two such cases for three planes in R 3. Can you find all the other cases?

11 EXERCISES Figure (a): Inconsistent Figure (b): Consistent If an equation is homogeneous (that is, the constant term is zero), then the associated geometric object passes through the origin. Let us revisit the solutions to homogeneous systems of equations. OBSERVATION. From a - (or 3-) dimensional point of view, we can think of observation 8 as follows. A homogenous system of m equations in (or 3) unknowns is just a system of m straight lines (or planes) all of which pass through the origin. Hence they all intersect (at least) at the origin, namely at the point (, ) (or (,, )). Higher-dimensional analogues exist, and follow a very similar train of thought. However, they are harder to visualize! Not impossible, though; just close your eyes... Exercises () Let s practice some basics first! For each of the following systems of linear equations, rowreduce to echelon form, identify the basic and free variables, and determine the rank. If the system is consistent, row-reduce to reduced echelon form and find all solutions. You may use the abbreviated notation if you want. (a) x y = 3 x y = (b) x 3y = 6x + 9y = 4 (c) x + y + 3z = y + z = x + y = 5 (d) u + v = 3 u + v = 3u v = 5 (e) x + y + 3z = x + y + 4z = x y + 3z = 3

12 . LINEAR EQUATIONS & THEIR SOLUTIONS (f) x y + z = 3 y + z = x y + 4z = (g) x 4y + z = 3x + y z = 5 (h) v + 4x y = u v + 5y 4z = u + x 3y + z = 3 (i) x + x 3x 3 x 4 + 4x 5 = x + 5x 8x 3 x 4 + 6x 5 = 4 x + 4x 7x 3 + 5x 4 + x 5 = 8 (j) x + x 3x 3 x 4 + 4x 5 = x + 5x 8x 3 x 4 + 6x 5 = x + 4x 7x 3 + 5x 4 + x 5 = () Compare the solutions to parts (i) and (j) above. Can you generalize that comparison to homogeneous vs. non-homogeneous systems of equations that share the same coefficients? (3) (a) Sketch the three straight lines x + y = x y = y = 3 (b) Can these three equations be solved simultaneously? (c) What happens to the figure if all right hand sides are zero? (d) Is there any nonzero choice of right hand sides which allows the three lines to intersect at the same point? (4) Consider the following system of equations in 3 variables: x + y z = 3 x y + z = (a) Find a third equation to include in the system so that there is a unique solution to all three equations. (b) Find a third equation which has no coefficients in common with the first two (try not to use the same equations as the first two with only modifications to the right-hand-side; make it interesting!) such that there are no solutions. (c) Find a third equation which has no coefficients in common with the first two (i.e. make it interesting again!) such that there are infinitely many solutions. Provide some justification in each of the three cases. (5) Find all values of c such that the following system of equations: (a) does not have any solutions. (b) has only one solution. (c) has infinitely many solutions. x + y = x + cz = x + cy + 8z = (6) Find all values of a and b such that the following system of equations: (a) does not have any solutions. (b) has only one solution. (c) has infinitely many solutions. x + 3y + z = x + 4y z = x + y + az = b (7) In this exercise, the polynomial coefficients become the unknowns! (a) Find a straight line which passes through the points (, ) and (3, ).

13 EXERCISES 3 (b) Find a parabola which passes through the points (, ) and (3, ). (c) Find a parabola which passes through the origin and points (, ) and (3, ). (d) Find a polynomial p(x)ofdegree3suchthatp() =, p() =, p(3) = 3 and p(4) = 4. (e) Exploration. Formulate a conjecture which generalizes these computations. (8) We continue to expand on the ideas in the previous exercise. (a) Find two polynomials p(x) =ax + b amd q(x) =cx + d whose sum equals x +and their difference p q equals 47x. (b) Consider the function f(x) = αsin(x)+ βcos(x) (this is called a linear combination of sin(x) andcos(x)). Find numbers α and β if f(π/4) = 3 and f(π/3) = 4. (c) Can you find numbers α and β such that f(x) =x +x+? If so, we say that x +x+ can be written as a linear combination of sin(x) andcos(x). (9) Using the lingo of the previous exercise, we say that a polynomial q(x) can be written as a linear combination of the polynomials p (x), p (x),,p k (x) if we can find numbers a, a,,a k such that a p (x)+a p (x)+ +a k p k (x)=q(x) For instance, any polynomial of degree 3 or less can be written as a linear combination of x 3, x, x and. Determine whether q(x) =x 3 x+4canbewrittenasalinear combination of each of the following sets of polynomials. (a) p (x) = x 3 p (x) = x 3 +x p 3 (x) = x 3 +x +x p 4 (x) = x 3 +x +x+ (b) p (x) = x 3 +x x p (x) = x 3 +x + p 3 (x) = x +3x+ p 4 (x) = x 3 +3x +x (c) p (x) = x 3 +x +x+ p (x) = x 3 +4x +3x () Equation Island is inhabited by chickens and foxes. Eighty percent of the chickens are hens, and each hen has chicks every year. Half the foxes are female, and each female has one pup every year. Additionally, each of the adult foxes (i.e. you don t have to take the newborns into account) eats chickens a year. Denote by f n and c n the number of foxes and chickens, respectively, alive on the island at the beginning of year n. (a) Find a formula for f n+ in terms of f n and c n.dothesameforc n+. (b) Find a formula of f n+ in terms of f n and c n.dothesameforc n+. (c) Can you find a general formula for f n+k in terms of f n and c n? How about for c n+k. () Consider the generic system of equations in two unknowns. ax + by = α cx + dy = β (a) Suppose ad bc. Show that this system has a unique solution. (Hint: divide your answer into two cases, depending on whether a equals zero or not.) (b) Suppose ad = bc. What can you say about the existence of solutions? () Linear equations can be thought of as bits of information: x y = is another way of saying that x equals y. If an equation is obtained from another by multiplying both sides by a number, then we can say that both equations contain the same information: x +y= saysthesamethingasx y=. We can further expand this notion by claiming, for instance, that the equation x y = can be described by the two equations x y = x + y = 6 since 3/5 times the first equation plus /5 times the second equation yields x y =.

14 4. LINEAR EQUATIONS & THEIR SOLUTIONS (a) How can you tell if one of the equations in a given system can be described by some of the other equations in that system? (b) Show that the last equation in the following system can be described using the equations before it. Express that description explicitly. x y + 3z = 3x + z = 6 x + y + 7z = 4 (3) Exploration. How do row operations manifest geometrically? Suppose you are dealing with a system of equations in unknowns, represented as a set of straight lines in R. If you switch two rows, that is akin to relabeling the corresponding lines. What happens if you multiply one row by a nonzero number? What happens if you add a multiple of one row to another? What can you say about reduced echelon form? (4) Consider the sequence,, 3, 5,,, 43, 85,. Let x n denote the nth term of this sequence; so x =,x =,x 3 =3,etc. (a) Write down a linear equation which defines x n in terms of previous two terms in the sequence. (b) Can you find a general formula for x n in terms of n? (In general, this is difficult without eigentheory). (c) Can you guess the limit of ratios of successive terms, x n /x n+,asn? Is that useful for finding an approximate formula for x n? (5) Most mathematical questions whose answers are no! can be reformulated with an eye towards more of a compromise. For instance, if a system of linear equations is inconsistent, perhaps we can find the next best thing to a solution. (a) Verify that the following system of equations is inconsistent. x + y = x y = x + y = 3 (b) Since we cannot find a solution to the above system, let us try to find solutions which make the left-hand-sides as close as possible to their respective right-hand-sides. More precisely, find numbers x and y such that the expression ( x + y +) +(x y ) +(x+y 3) is as small as possible. (c) What would happen if we used the same approach on a consistent system of linear equations? (6) Proofs. Determine if each of the following statements is true. If it is, provide a proof. If it is false, demonstrate that with a counterexample, and change the statement in a minimal way to make it true. (a) If x = x,y = y and x = x,y = y are two solutions to the equation ax + by = c, then so is x = x + x, y = y + y. In other words, the sum of solutions is also a solution. (b) Let r be the rank of a system of m equations in n unknowns. (i) If r = m, then the system must be consistent. (ii) If r = n, then the system can have at most one solution.