2 Solve: 10x4 - 9x2 + 2 = 0
Jun 26, 2011 · c) ther is no solution a) solution set is {x -2 < x < -1, x > 10 } 38 Add Simplify if possible 2y + y y2-16 y-4 2y y2"16 + y y"4 = y(y+6) y2"16 39 For the following equation, state the value of the discriminant and then describe the nature of the solution: 15x2 + 5x – 14 = 0 The value of discriminant is: a> The equation has one real
10X Cell Dissociation Solution - Trevigen
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Dako Target Retrieval Solution (10x) ENGLISH Code S1699
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SOLVING SOLUTION AND MIXTURE VERBAL PROBLEMS 1 Solution Problems
7 An 8 solution of local anesthesia is needed However, 40 fl oz of a 10 solution is the only type of this anesthesia in stock How much neutral solution containing no anesthesia should be added to the 40 fl oz container of the 10 solution? 8 A pharmacist needs to fill an order for a 4 lidocaine solution topical However, only 2 and 9
Math 2331 { Linear Algebra
Parallel solution sets of Ax = b and Ax = 0 Theorem Suppose the equation Ax = b is consistent for some given b, and let p be a solution Then the solution set of Ax = b is the set of all vectors of the form w = p+ v h, where v is any solution of the homogeneous equation Ax = 0 Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 12
Solution The information we are given can be summarized this
Solution The information we are given can be summarized this way dG ds =0 05gal/mi, ds dt =30mi/hr, and we want to find this: dG dt =? If we give these numbers and units the usual interpretation, we have this: • We use 0 05 gallons by driving one mile, • We drive 30 miles in one hour, • How many gallons will we use in one hour?
13 Congruences - NIU
a solution if an only if gcd(a,n) is a divisor of b Also note that the congruence is stated modulo 90, and so the most satisfying answer is given in terms of congruence classes modulo 90 Solution: We have gcd(42,90) = 6, so there is a solution since 6 is a factor of 12 Solving the congruence 42x ≡ 12 (mod 90) is equivalent to solving the
1 Calculate the pH of a 10 x 10-3 M solution of
8 Calculate the pH and pOH of a solution that was made by adding 400 ml of water to 350 ml of a 5 0 x 10-3 M NaOH solution 5 Calculate the pH of a 2 0 liter solution containing 0 005 g of HCl 6 Calculate the pH and pOH of a solution with a volume of 5 4 liters that contains 15 g of hydrochloric acid (HCl) and 25 g of nitric acid (HNO 3) 7
1 In T(n) = a * T(n/b) + f(n) , a refers to
10 In the matrix chain product of (2x10)(10X4)(4X7) the solution is A) 136 B) 420 C) 360 D) 128 11 In a multistage graph the minimum no of stages are A) 0 B)3 C) 1 D)2 12 One of the methods for All Pairs shortest path problem is similar or equivalent to
MATH 351 Solutions
Solution This cannot be a probability density function If c= 0, then it does not integrate 1 For any c6= 0, there is an interval in −2 ≤ x≤ 2 over which the integral is negative, and therefore does not represent a probability over this interval 2 Suppose that f(x) = (c(3x−x2) if 0 ≤ x≤ 2 0 otherwise
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1.5 Solutions Sets of Linear Systems
Math 2331 { Linear Algebra
1.5 Solutions Sets of Linear Systems
Jiwen He
Department of Mathematics, University of Houston
jiwenhe@math.uh.edu math.uh.edu/jiwenhe/math2331Jiwen He, University of HoustonMath 2331, Linear Algebra1 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
1.5 Solutions Sets of Linear SystemsHomogeneous System
Nontrivial Solutions
Writing Solution Set in Parametric Vector Form
Nonhomogeneous System
Theorem: Solutions of Nonhomogeneous System
Writing Solution Set in Parametric Vector Form
Jiwen He, University of HoustonMath 2331, Linear Algebra2 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Homogeneous System
Homogeneous System
Ax=0 (Aismnand0is the zero vector inRm)Example x1+ 10x2= 0
2x1+ 20x2= 0
Corresponding matrix equationAx=0:
1 10 2 20 x1 x 2 =0 0Trivial solution: x=0
0 orx=0Jiwen He, University of HoustonMath 2331, Linear Algebra3 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Homogeneous System: Nontrivial Solutions
The homogeneous systemAx=0alwayshas thetrivial solution, x=0.Nontrivial Solution Nonzero vector solutions are callednontrivial solutions.Example (cont.)Donontrivialsolutions exist?1 10 0
2 20 0
1 10 0
0 0 0Consistent system with a free variable has innitely many solutions.A homogeneous equationAx=0has nontrivial solutions if
and only if the system of equations has. Jiwen He, University of HoustonMath 2331, Linear Algebra4 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Homogeneous System: Example 1
Example (1)
Determine if the following homogeneous system has nontrivial solutions and then describe the solution set.2x1+ 4x26x3= 0
4x1+ 8x210x3= 0Solution:There is at least one free variable (why?)
=)nontrivial solutions exist2 46 0
4 810 0
s1 23 04 810 0
s1 23 00 0 2 0
s1 2 0 0
0 0 1 0
=)x 1= x2is free
x3=Jiwen He, University of HoustonMath 2331, Linear Algebra5 / 12
1.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Homogeneous System: Example 1 (cont.)
x=2 4x 1 x 2 x 335 =2 42x2
x 2 03 5 =2 42
1 03 5 =x2v
Graphical representation:
solution set = spanfvg= line through0inR3Jiwen He, University of HoustonMath 2331, Linear Algebra6 / 12
1.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Example 2
Example (2)
Describe the solution set of
2x1+ 4x26x3= 0
4x1+ 8x210x3= 4
( same left side as in the previous example)Solution:2 46 0
4 810 4
row reduces to1 2 0 60 0 1 2
x=2 4x 1 x 2 x 335 =Jiwen He, University of HoustonMath 2331, Linear Algebra7 / 12
1.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Example 2 (cont.)
x=2 460 23
5 +x22 42
1 03 5 =p+x2v
Graphical representation:
Parallel solution sets ofAx=0&Ax=bJiwen He, University of HoustonMath 2331, Linear Algebra8 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Recap of Previous Two
ExamplesExample (1. Solution ofAx=0)x=x22
421 03 5 =x2v x=x2v= parametric equation of line passing through0andvExample (2. Solution ofAx=b)x=2 46
0 23
5 +x22 42
1 03 5 =p+x2v x=p+x2v= parametric equation of line passing throughp parallel tovJiwen He, University of HoustonMath 2331, Linear Algebra9 / 12
1.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Theorem
Parallel solution sets of
Ax=bandAx=0Theorem
Suppose the equationAx=bis consistent for some givenb, and letpbe a solution. Then the solution set ofAx=bis the set of all vectors of the formw=p+vh, wherevhis any solution of the homogeneous equationAx=0.Jiwen He, University of HoustonMath 2331, Linear Algebra10 / 121.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Example
Example
Describe the solution set of 2x14x24x3= 0; compare it to the solution set 2x14x24x3= 6:Solution:Corresponding augmented matrix to 2x14x24x3= 0:244 0(ll-in)
Vector form of the solution:
v=242x2+ 2x3
x 2 x 335 =2 42
1 03 5 +2 42
0 13 5
Corresponding augmented matrix to 2x14x24x3= 6:
244 6(ll -in)Jiwen He, University of HoustonMath 2331, Linear Algebra11 / 12
1.5 Solutions Sets of Linear SystemsHomogeneousNonhomogeneous
Nonhomogeneous System: Example (cont.)
Vector form of the solution:
v=243 + 2x2+ 2x3
x 2 x 335 =2 43
5 +2 42
1 03 5 +2 42
0 13