College Algebra

Main topic: Linear Equations and Inequalities in One Variable Subtopic: Solve applied problems involving linear equations    John has $9.60 in pennies and nickels. If there are thrice as many nickels as pennies, how many pennies does John have? How many nickels does he have? A. 360 Pennies and 120 nickels B. 120 Pennies and 360 nickels C. 60 Pennies and 180 nickels D. 180 Pennies and 60 nickels Answer: C Rationale Let p be the number of pennies. There are thrice as many nickels as pennies. So, there are 3p nickels. First step is converting all the currencies in the same unit – ‘cent’. As $1 = 100 cents, $ 9.60 = 960 cents As 1 penny = 1 cent, p pennies = p cents As 1 nickel = 5 cents, 3p nickels = 15p cents Second step is constructing an equation using given information. Number of cents for pennies + number of cents for nickels = Total number of cents. So equation is,  p + 15p = 960 After simplifying left side, 16p = 960 Or p = 60 Therefore, number of pennies = p = 60 and number of nickels = 3p = 3*60 = 180. Answer option A is incorrect. Check the given relation between pennies and nickel. The number of nickels is three times the number of pennies. Answer option B is incorrect. Check the given relation between pennies and nickel. The number of nickels is three times the number of pennies. Answer option D is incorrect. Check the given relation between pennies and nickel in the final step.

Main Topic: Graphing Polynomial and Rational Functions  Subtopic: Use the technique of completing the square to interpret a quadratic function as a transformation of x2 Which transformation is needed to be used on x2, to get the graph of f(x) = 2x2 – 12x + 22? A: Shift right by 3 units, stretch vertically by a factor 2 and then shift upward by 13 units B: Shift left by 3 units, stretch vertically by a factor 2 and then shift upward by 4 units C: Shift right by 3 units, stretch vertically by a factor 2 and then shift upward by 4 units - Correct option D: Shift right by 3 units and shift upwards by 4 units Answer: C Rationale First step is to use the technique of completing a square. 2x2 – 12x + 22 = 2 (x2 – 6x) + 22 = 2 (x2 – 6x + 9 - 9) + 22 = 2 (x2 – 6x + 9) – 18 + 22 = 2 (x – 3)2 + 4 Next step is to decide the transformation on x2. First transformation is: shift graph of x2 to right side by 3 units to get a graph of (x - 3)2. Then stretch the graph of (x - 3)2 vertically by factor 2 to get a graph of 2(x - 3)2. Finally shift the graph of 2(x - 3)2 upwards by 4 units to get a graph of 2(x - 3)2 + 4. Answer option A is incorrect. Check your calculations for completing the square method. Answer option B is incorrect. Check your calculations for completing the square method. Also check the transformation from f(x) to f(x - c). Answer option D is incorrect. Check your calculations for completing the square method. Also check the transformation from f(x) to c*f(x).

Main Topic: Exponential and Logarithmic Functions    Subtopic: Convert expressions from exponential notation to logarithmic notation and vice versa What is the result after converting the logarithmic expression   into exponential form? A: x2 = 28 - Correct option B: x2 = 108 C: (x2)8 = 2 D: x2 = 82 Answer: A Rationale If loga b = c, its exponential form is written as b = ac. Applying this to the original problem   , the exponential form becomes x2 = 28 Therefore the required answer is: x2 = 28 Answer option B is incorrect. Base of logarithmic function is 2, not 10. Answer option C is incorrect. Recall conversion of logarithmic function into exponential function as, the exponential form of logab = c is b = ac. Answer option D is incorrect. Base of logarithmic function is 2, not 8.    

Main Topic – Exponential and Logarithmic Equations    Sub topic - Solve logarithmic equations Solve the equation log(x) + log(x + 4) = log (32) for x A. x = -8 B. x = 4  C. x = 4 and x = -8 D. x = 14 Answer: B Rationale Rules of Logarithm are used. First step is to use the addition rule of logarithm. log (a) + log (b) = log (a*b) Therefore, the equation becomes log [x (x + 4)] = log(32) Second step is to get rid of logarithm from both sides. So, x (x + 4) = 32 => x2 + 4x = 32 Third step is, to solve this quadratic equation. x2  + 4x – 32 =0 (x - 4) (x + 8) = 0 x = 4 and x = -8 Next step is, to identify the correct solution. x = -8 cannot be the solution, since log(x) is not defined for negative values. Therefore x = 4 is the only solution. Answer option A is incorrect: log(x) is not defined for negative values. Answer option C is incorrect: log(x) is not defined for negative values. Answer option D is incorrect: Use the addition rule of logarithm and then solve the quadratic equation by using factor method or quadratic formula.

Main Topic – System of Equations Sub Topic - Determine whether a system of linear equations has no solutions (inconsistent system), one solution, or infinitely many solutions (dependent system). Determine the type of solution for given system of equations. 2x + 3y + 4z = 9 3x + 5y -2z = 6 7x + 4y + z = 12 A. No solutions B. Three solutions C. Infinitely many solution D. One solution Answer: D Rationale First step is, to eliminate one variable from the first two equations. Let us eliminate z first from the first two equations. To eliminate z, multiply second equation by 2 and add it in the first equation.    2x + 3y + 4z = 9 + 6x + 10y - 4z = 12 ----------------------------    8x + 13y        = 21 Second step is to eliminate z from the last two equations. Multiply third equation by 2 and then add it in the second equation.      3x + 5y - 2z = 6 + 14x + 8y + 2z = 24     -------------------------     17x + 13y      = 30 Next step is, to solve the system of equation with two variables which we obtained by eliminating z. 8x + 13y = 21 17x + 13y = 30 Elimination method is used:    8x + 13y =21  17x + 13y = 30   -       -          - ------------------------    -9x           = -9 So, x = 1, Use this value of x in any of the above equations. 8x + 13y = 21 => 8*1 + 13y = 21 => 13y = 13 So y = 1 Now use the value of x and y in any of the original equation and solve for z. 2x + 3y + 4z = 9 => 2*1 + 3*1 + 4z = 9 => 5 + 4z = 9 => 4z = 4 => z = 1 Therefore the solution of system is x = 1, y = 1 and z = 1. Since it has a unique solution, the type of solution is one solution. Answer option A is incorrect. Eliminate z variable from the first two and the last two equations. Then solve the remaining system of equations with two variables by using the elimination method. Answer option B is incorrect. Number of variables in the system of equations is three but the number of solutions is not the same as the number of variables. Answer option C is incorrect. Eliminate z variable from the first two and the last two equations. Then solve the remaining system of equations with two variables by using the elimination method.