Math 20 Practice Exam #2 Problems and Their Solutions! #1) Solve the linear system by graphing: Isolate for in both equations. Graph the two lines using the slope-intercept method. The two lines intersect at the point #2) Solve the linear system by graphing: Graph the two lines using the slope-intercept method. The two lines are parallel because they have the same slope but different y-intercepts. The system has no solution! The answer can also be expressed as. #3) Solve the linear system by graphing: Graph As the two equations are identical, the lines coincide and the points on the line are the solutions. Thus, the system has infinitely many solutions. The solution set consists of all points lying on either coinciding line The solution set can be expressed in two equivalent ways: #4) Graph the solution set of the linear inequality:.).) Graph the equation and draw a solid line indicating that the line is a part of the solution. Then below the line based on the fact that is isolated and the symbol is. Use the to verify the is correct. #5) Page 166 #29 in the book...graph the solution set of the system of linear inequalities:
#6) Graph the linear inequality. The graph is of two lines with the area between them shaded. The graph of the equation should be a broken line and the graph of the line should be a solid line. #7) Solve the system using the addition method. I) II) III) The solution is #8) Solve the system by the addition method. I) Because both variables are eliminated and we have a false statement, this system has no solution. #9) Solve the system by the substitution method. I) II) III) The solution is. #10) Solve the system by the substitution method. I) Because both variables are eliminated and we have a true statement the system has infinitely many solutions. The solution set are the points on the coinciding lines so we can write the answer as #11) Solve the system by either addition or substitution. To begin, clear the fractions so that the coefficients are integers. I II)
III) The solution is #12) A person invests $6,000 in two stocks paying 9% and 6% annual interest, respectively. At the end of the year, the total interest from these investments was $480. How much was invested at each rate # % # % I) II) III) $4,000 was invested at 9% and 2,000 was invested at 6%. #13) A chemist needs to mix a 20% alcohol solution with a 50% alcohol solution to obtain 60 ounces of a 30% alcohol solution. How many of each of the solutions must he use % % I) II.) 40 ounces of 20% alcohol solution need to be mixed with 20 ounces of a 50% alcohol solution to obtain 60 ounces of a 30% alcohol solution. #14) When a plane flies with the wind, it can travel 4,200 miles in 6 hours. When the plane flies in the opposite direction, against the wind, it takes 7 hours to fly the same distance. Find the speed of the plane and the speed of the wind. I) II) #15) A company is planning to produce and sell a new line of computers. The fixed cost will be $360,000 and it will cost $850 to produce each computer. Each computer will be sold for $1,150. Find the cost function of producing computers, the revenue function of selling computers, and the break-even point. Describe what the breakpoint means in these circumstances.
, $ #16) Solve by eliminating variables using the addition method. The three planes intersect at #17) Solve using matrices. The lines intersect at #18) Solve using matrices.
Start with equation #3) Continue with equation #2) Finish with equation #1) The three planes intersect at. #19) Solve using matrices. #20) Solve and graph. The graph has a bracket at and is shaded to the left on a number line. The answer is interval notation is The answer is set builder notation is #21) A student has scores of 70, 79, 85 and 88 on four exams. What score does she need on the 5th exam to have her average be 80 or more The student needs a score of 78 or more on the fifth exam. 22) Solve and graph the compound inequality.
The graph has a parentheses at and a bracket at and is shaded between those two numbers on a number line. The answer in interval notation is The answer in set builder notation is #23) Solve and graph the compound inequality and. The intersection of the two sets is all the points on the number line that the two sets have in common so the solution is The graph has a parentheses at and includes all the points to the left of The answer in interval notation is The answer in set builder notation is #24) Solve the compound inequality or #25) Solve #26) Solve #27) Solve and graph. The graph has brackets at and and is shaded in between. The solution in interval notation is [ The solution in set builder notation is #28) Solve and graph. The graph is shaded from negative infinity to 1 where a parentheses needs to be placed. Then there is a parentheses at and the shading continues to infinity. In interval notation the solution is In set builder notation the solution is #29) Solve and graph. The rules for inequalities with absolute values do not apply because the number on the right side of the inequality is negative. Since the absolute value of any number cannot be negative, and because any positive number or 0 is greater than 2 the inequality is true for all values of The solution set is the interval (, ) and the graph is all real numbers on the number line.
#30) Solve and graph. The rules for inequalities with absolute values do not apply because the number on the right side of the inequality is negative. This is an impossible inequality because no matter what values we substitute for we get no solution whatsoever. Because the absolute value of a number is either positive or 0, it cannot be less than or equal to This inequality has NO SOLUTION! #31) a) Find the intersection: {red, white, blue} {red, green, blue} {red, white, blue} = {red, blue} b) Find the union: {red, white, blue} {red, green, blue} {red, green, blue} {red, white, blue} {red, green, blue}= {red, white, blue, green} Graph the line by plotting the intercepts and Graph the vertical lines and The vertices of the common region are, d) Evaluate the objective function for total weekly earnings at each of the 4 corners of the graphed region. Corners Objective Function $ $ $ $ e) At what corner does the maximum value of the objective function occur What does it tell us The maximum value occurs at the corner. Given the constraints in the problem, it means that the student will make the most money ($164) if he or she tutors for 8 hours and serves as an aide for 12 hours on a weekly basis. 32) a) A student earns $10 per hour for tutoring and $7 per hour as a teacher's aide. Let the number of hours of tutoring each week and the number of hours as a teacher's aide each week. Write the objective function that describes weekly earnings. b) The student is bounded by the following constraints: To have enough time to study, the student can work no more than 20 hours a week. The tutoring center requires that each tutor spend at least 3 hours a week tutoring. The tutoring center requires that each tutor spend no more than 8 hours a week tutoring. c) Graph the system of inequalities above. As and must be nonnegative, use only the first quadrant.