8 Systems of Linear Equations

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1 8 Systems of Linear Equations 8.1 Systems of linear equations in two variables To solve a system of linear equations of the form { a1 x + b 1 y = c 1 x + y = c 2 means to find all its solutions (all pairs (x, y) which check both equations). Methods for solving such a system: a) Graphical: graph both linear equations and find the point(s) of intersection. Then read the solution from the graph. b) Elimination: multiply each equation by convenient numbers and add them, such that one of the variables will be eliminated. Once you have found one unknown, replace its value in the equation to find the other one. c) Substitution: solve one equation for a variable, and replace this value in the other equation. Solve (find one of the unknowns) and then replace its value in the equation to find the other one. Remark: the system above has: i) one solution (a pair of values for x and y). This happens when the lines intersect at one point: ii) no solution (the equations are inconsistent). This happens when lines are parallel and distinct: a 1 b 1 a 1 = b 1 c 1 c 2 iii) infinitely many solutions (the equations are dependent). This happens when the lines coincide: a 1 = b 1 = c 1 c 2. Exercise { 1 Solve by one of the methods { above: 3x 2y = 6 4x + 3y = 10 a) b) 2x + 4y = 20 2x + y = 4 { { 4x 3y = 2 2x 3y = 6 e) f) g) 8x 6y = 4 y = 3x 5 c) { 2x + 3y = 5 x 2y = Systems of linear equations in three variables To solve a system of linear equations of the form a 1 x + b 1 y + c 1 z = d 1 x + y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 { 3x 5y = 2 2x 3y = 1 means to find all its solutions (all triples (x, y, z) which check both equations). Methods for solving such a system: a) Elimination: eliminate a variable from the three equation d) { 5x 2y = 1 10x + 4y = 3 b) Substitution: solve one equation for one of the variables, and replace the value in the other two equations. Solve the resulting system (two equations, two unknowns), then find the value of the remaining unknown. c) Linear algebra: (Cramer s rule) Remark: when solving by either method your variables might disappear. You may obtain something like 0 = 0 (equations are dependent, infinitely many solution), or 0 = 1 (equations are inconsistent, no solution). Exercise 2 Solve the given systems: x + y + z = 6 a) 2x y + z = 3 b) x + 2y 3z = 4 x 5y + 4z = 8 d) 3x + y 2z = 7 e) 9x 3y + 6z = 5 2x + y z = 3 3x + 4y + z = 6 2x 3y + z = 1 x + 3y = 5 6y + z = 12 x 2z = 10 2x + 3y z = 5 c) 4x + 6y 2z = 10 x 4y + 3z = 5 1

2 8.3 Determinants (briefly) A 2 2 determinant: A 3 3 determinant a 1 b 1 c 1 c 2 a 3 b 3 c 3 a c b d = ad bc = a 1 c 3 + b 3 c 1 + a 3 b 1 c 2 a 3 c 1 b 1 c 3 a 1 b 3 c 2 Remark: to help you remember the above definition, use triangle rule or Sarrus rule. Exercise 3 Compute the given determinants: a) b) c) 3 x 2 x d) e) Remark: there are other rules for computing determinants, for example expansion by a row (or a column). What is that? 8.4 Cramer s rule To solve compute the determinants: a 1 b 1 c 1 D = c 2 a 3 b 3 c 3, D x = d 1 b 1 c 1 d 2 c 2 d 3 b 3 c 3 a 1 x + b 1 y + c 1 z = d 1 x + y + c 2 z = d 2, a 3 x + b 3 y + c 3 z = d 3, D y = Cramer s rule says that if D 0, the solution of the system is given by a 1 d 1 c 1 d 2 c 2 a 3 d 3 c 3, D z = a 1 b 1 d 1 d 2 a 3 b 3 d 3. Can you generalize to four unknowns? Exercise 4 Solve using Cramer s rule: x = D x D, x + y = 1 2x z = 3 y + 2z = 1 y = D y D,. z = D z D. 8.5 Word problems Same as before: set the notations, write equations (system of equations), solve, and then answer the question Exercise 5 One number is 2 more than 3 times another, and their sum is 26. Find the two numbers. Exercise 6 Suppose 850 tickets were sold for the game for a total of $1, 100. If adult tickets cost $1.50 and children s tickets cost $1.00, how many of each kind of ticket were sold? Exercise 7 Suppose a person invests a total of $10, 000 in two accounts. One account earns 5% annually and the other earns 6% annually. If the total interest earned from both accounts in a year is $560, how much is invested in each account? Exercise 8 How much 20% alcohol solution and 50% alcohol solution must be mixed to get 12 gallons of 30% alcohol solution? Exercise 9 It takes 2 hours for a boat to travel 28 miles downstream. The same boat can travel 18 miles upstream in 3 hours. What is the speed of the boat in still water and the speed of the current of the river? 2

3 9 The conic sections 9.1 The parabola A parabola is the graph of a quadratic of the form y = ax 2 + bx + c (a 0). To sketch its graph, use the following: a) if a > 0, the parabola opens upward, and downward if a < 0 b) the intercepts are: y int = c and x int = b ± 4ac 2a ( c) the vertex has coordinates b ) 2a, 4ac b2 4a (if = 4ac < 0 there are no x-intercepts!) Exercise 10 Sketch the graph of: a) y = x 2 4x 5 b) y = x 2 4x + 12 c) y = 3x 2 6x + 1 d) y = 2x 2 + 6x 5. Exercise 11 Graph the inequality y x The circle Recall that the distance between the points (x 1, y 1 ) and (x 2, y 2 ) is dist = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Exercise 12 Find the distance between (3, 5) and (2, 1). Exercise 13 Find x if the distance from (x, 5) to (3, 4) is 2. The equation of a circle of radius r > 0 centered at (a, b) is (x a) 2 + (y b) 2 = r 2. Exercise 14 Find the equation of the circle with center at ( 3, 2) having a radius of 5. Exercise 15 Give the equation of the circle with radius 3 whose center is at the origin. Exercise 16 Find the center and radius, then sketch the graph of the circle whose equation is (x 1) 2 +(y+3) 2 = 4. Exercise 17 Sketch the graph of x 2 + y 2 + 6x 4y 12 = 0. (Hint: complete the square) Exercise 18 Graph x 2 + y 2 < Functions A function is a rule f which assigns to each x in a set A (called domain) a unique number y in a set B (called range). We write f : A B, y = f (x). To graph a function f : A B means to graph all points (x, f (x)) for x A. Examples: a) linear function: f : R R, f (x) = ax + b b) quadratic function: f : R R, f (x) = ax 2 + bx + c c) exponential function: f : R R, f (x) = a x (a > 0 - why?) d) trigonometric functions: f : R R, f (x) = sin x (or cos x, tan x, cot x) 3

4 Exercise 19 Graph the given functions: a) f : R R, f (x) = 2x + 3 b) g : R R, g (x) = 2x 2 + 3x 5 c) h : R R, h (x) = 2 x d) k : R R, k (x) = ( 1 2) x. Exercise 20 What happens in the exercise above if we change the domain of the functions to: a) [0, ) b) (, 0) c) ( 1, 1]? Composition of functions: if g : A B and f : B C, the composition of f and g is the function f g : A C defined by g (x) = f (g (x)). Exercise 21 Find the composition of the functions f, g : R R, given by f (x) = 2x + 3 and g (x) = x Are they equal? Exercise 22 Given two functions f : A B and g : C D, when can you compose f g? g f? Inverse function: A function f : A B has an inverse, if there is a function g : B A such that g f (x) = x for all x A AND f g (x) = x for all x B. Notation: the inverse function of f is denoted f 1 (not to be confused with 1 f!). Exercise 23 Verify that the inverse of the function f : R R, f (x) = x 3 is the function g : R R, g (x) = 3 x. Exercise 24 Does the function f (x) = x 2 have an inverse? What is its domain and range? Remark: the function and its inverse undo what the other one did. Question: how do we know a function has an inverse? Answer: if the function is bijective (definition below). A function f : A B is: a) injective if whenever x 1 x 2 (x 1,2 A), we have f (x 1 ) f (x 2 ) b) surjective if for each y in B we can find a x A such that f (x) = y c) bijective if it is both injective and surjective. Remark: an alternate way to see if a function f : A B is injective/surjective is by looking at its graph (horizontal line test): a) a function is injective if a horizontal line intersects the graph at most once b) a function is surjective if a horizontal line (y = b, b B) intersects the graph at least once Question: if a function f : A B has an inverse, how can we find it? Answer: solve f (x) = y for x. The resulting value for x is the expression for f 1 (y). Exercise 25 Find the inverse of the function f : R R, f (x) = 2x Logarithms Exercise: Graph: a) y = 2 x b) y = ( 1 2) x. Remark: the exponential function f : R (0, ), f (x) = a x (with a > 0, a 1) has an inverse (both injective and surjective). We denote by log a x its inverse (g : (0, ) R, g (x) = log a x). What this means: log a b = c iff a c = b. Exercise 26 Find the indicated logarithms: log 2 8, log 5 25, log , log Exercise 27 Solve the given equations: a) log 3 x = 2 b) log x 4 = 3 c) log 8 4 = x Notation: decimal logarithm lg x = log 10 x and natural logarithm ln x = log e x (where e ) 4

5 11.1 Properties of logarithms Assuming x, y > 0 and a, b > 0 and a, b 1, we have (why?): 1. log a (xy) = log a x + log a y (logarithm of a product) ( ) x 2. log a = log y a x log a y (logarithm of a ratio) 3. log a x r = r log a x (logarithm of a power) 4. log a x = log b x log a x (change of base formula) Exercise 28 Expand using the properties of logarithm: a) log 5 3xy z b) log 2 x 4 yz 3 Exercise 29 Convert to a base decimal logarithm: a) log 8 24 b) log Exercise 30 Solve for x: log 2 (x + 2) + log 2 x = Exponential equations Idea: to solve, take logarithms (with a convenient base) of both sides. Exercise 31 Solve: a) 5 x = 12 b) 25 2x+1 = 12 c) 6 5 2x = 4. 5