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All Textbook Solutions for Precalculus
For Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. v+v=0For Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. cv+w=cv+cw51PEFor Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. cdv=cdv53PEFor Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. cv=cv55PE56PE57PEFor Exercises 55-60, find a unit vector in the direction of v. (See Example 5) s=3.9i+5.2j59PE60PEFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j v+sFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j w+rFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j s-rFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j v-wFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j 2r+3wFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j 6s-4vFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j r+wFor Exercises 61-68, perform the indicated operations for the given vectors. (See Example 6) v=3.3i+11jw=4i+js=7.2i5.4jr=12i5j rwFor Exercises 69-74.write the vector v in the form ai+bj , where v has the given magnitude and direction angle. v=12,=3070PE71PE72PE73PE74PE75PE76PE77PE78PE79PE80PE81PE82PEA punter kicks a football with an initial velocity given by v=22.2i+18.6j m/sec. (See Example 8) a. Find the magnitude of the velocity vector at the time the ball leaves the punter's foot. Round to the nearest m/sec. b. Find the angle from the horizontal at which the ball was kicked. Round to the nearest degree.The velocity of a ship is given by the vector 6.4i+7.7j mph. a. Find the speed of the ship. Round to the nearest mph. b. Find the bearing of the ship. Round to the nearest degree.85PEA plane flies at a speed of 370 mph on a bearing of S45 E. Relative to the ground, the plane's speed is measured as 345 mph with a true bearing of S40 E. Rounding to the nearest whole unit, a. Express the velocity of the plane vp relative to the air in terms of i and j. b. Express the true velocity of the plane vT in terms of i and j. c. Express the velocity of the wind vw in terms of i and j and find the speed of the wind.A swimmer swims 1.5 mph in still water and wants to swim to a point e due north from her starting point directly across a river, if the current is 0.5 mph due east, at what angle 9 should the swimmer swim to reach point B1 Round to the nearest tenth of a degree. (See Example 10)A boat that travels 6 mph in still water leaves a marina in Miami to travel to an island called Elliott Key due south of the marina. The prevailing current is directed northeast N45E at 0.8 mph. At what bearing should the captain of the boat steer the ship?For Exercises 89-94, the given forces (in units of pounds) act on an object. a. Find the resultant force, R. b. What additional force F is needed for the object to be in static equilibrium? (See Example 11) F1=3,11andF2=8,690PE91PE92PEFor Exercises 89-94, the given forces (in units of pounds) act on an object. a. Find the resultant force, R. b. What additional force F is needed for the object to be in static equilibrium? (See Example 11)For Exercises 89-94, the given forces (in units of pounds) act on an object. a. Find the resultant force, R. b. What additional force F is needed for the object to be in static equilibrium? (See Example 11)95PE96PEFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u 2utFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u utFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u wvFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u v+uFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u v+wFor Exercises 97-102, refer to vectors t, u, v, and w in the figure and match each vector with an equivalent vector a-f. a.tb.3uc.wd.u2te.vf.3u vwA kayaker paddles 3 mph in still water. If she heads due east but encounters a 0.8 -mph current flowing south, a. Find the actual direction of travel. Write the direction as a bearing and round to the nearest degree. b. How far will she travel in 2 hr? Round to the nearest tenth of a mile.A long-distance swimmer swims 2 mph in still water, If he heads north but encounters a 0.6 -mph current flowing northeast, a. Find the actual direction of travel. Write the direction as a bearing and round to the nearest degree. b. How far will he swim in 1.5 hr? Round to the nearest tenth of a mile.105PEFor Exercises 105-106, two vectors v and w act on a point in the plane with the indicated force. a. Use a parallelogram to sketch the resultant vectorv+w , b. Use the law of cosines to find the magnitude of the resultant vector. [Hint: Adjacent angles in a parallelogram are supplementary.) c. Use the law of sines to find the angle that v+wa makes with v. Round answers to the nearest whole unit.107PEExplain how the sum v+w is found geometrically.109PE110PEFor Exercises 111-112, assume that the system is in equilibrium with F2 acting directly downward. Calculate the magnitude of the forces F1 and F2 . Round to the nearest tenth of a pound.For Exercises 111-112, assume that the system is in equilibrium with F2 acting directly downward. Calculate the magnitude of the forces F1 and F2 . Round to the nearest tenth of a pound.A 50 -lb box is supported by two ropes attached to the ceiling. Find the magnitudes of the tension vectors T1 , and T2 . Round to the nearest tenth of a pound. (See Example 12)A tightrope walker and the balance bar together weigh 175 lb. Find the magnitudes of the tension vectors T1 , and T2 . Round to the nearest tenth of a pound.Given v=2,6 andw=3,1 , find a. vw b. wv c.vv d. v2SPDetermine if the given vectors are orthogonal, a. v=4i+3j and w=6i10j b.s=2i+j and t=11i+22jRepeat Example 4 with v=7,3 and w=2,6 .Repeat Example 5 with a 1400 -lb SUV parked on a hill with an incline of 12 .6SP7SPThe dot product of v = a1,b1 and w = a2,b2 equals and the result is a (scalar/vector).If is the angle between two nonzero vectors v and w, then the cosine of can be found from the dot product of v and as cos= .3PETwo vectors are said to be if and only if they meet at a right angle.The projection of a nonzero vector v onto a nonzero vector w is given by projwv= .Suppose that D is the displacement vector of an object when the object is moved in a straight line from points A to B under a constant force F. Then the work done is given by W= .For Exercises 7-14.for the given vectors, find a. vw b. vv c. ww (See Example 1) v=3,5,w=10,8For Exercises 7-14.for the given vectors, find a. vw b. vv c. ww (See Example 1) v=6,7,w=1,49PEFor Exercises 7-14. for the given vectors, find a. vw b. vv c. ww (See Example 1) v=13,45,w=13,5811PEFor Exercises 7-14. for the given vectors, find a. vw b. vv c. ww (See Example 1) v=2j,w=14iFor Exercises 7-14. for the given vectors, find a. vw b. vv c. ww (See Example 1) v=2.1i6.8j,w=0.4i0.3jFor Exercises 7-14. for the given vectors, find a. vw b. vv c. ww (See Example 1) v=0.5i+1.6j,w=0.1i+2.3jFor Exercises 15-18, use the dot product to find the magnitude of the vector. v=4i8j16PE17PE18PE19PE20PE21PE22PE23PE24PE25PE26PE27PE28PEFor Exercises 29-32, let v=a1,b1,w=a2,b2,u=a3,b3 , and c be a real number. Prove the given statement. cvw=cvwFor Exercises 29-32, let v=a1,b1,w=a2,b2,u=a3,b3 , and c be a real number. Prove the given statement. 0v=031PEFor Exercises 29-32, let v=a1,b1,w=a2,b2,u=a3,b3 , and c be a real number. Prove the given statement. vw+u=vw+vu33PE34PEFor Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=8i+6j,w=4i+18jFor Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=12i5j,w=3i4jFor Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=1,7,w=12,8For Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=8,3,w=4,3For Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=cos45i+sin45j and w=cos60i+sin60jFor Exercises 35-40, find the angle between v and w. If necessary, round to the nearest tenth of a degree. (See Example 2) v=cos150i+sin150j and w=cos225i+sin225jIf v=10 and w=6 and the angle between v and w is 30 , find vw .If v=2.6 and w=8 and the angle between v and w is 60 , find vw .43PEFor Exercises 43-50, determine if the given vectors are orthogonal, parallel, or neither. (See Example 3) t=6,18,s=9,3For Exercises 43-50, determine if the given vectors are orthogonal, parallel, or neither. (See Example 3) w=15i14j,v=154i+3j46PEFor Exercises 43-50, determine if the given vectors are orthogonal, parallel, or neither. (See Example 3) p=2,6,q=-1,-3For Exercises 43-50, determine if the given vectors are orthogonal, parallel, or neither. (See Example 3) k=-32,2,s=6,-849PEFor Exercises 43-50, determine if the given vectors are orthogonal, parallel, or neither. (See Example 3) z=14i+7j,t=7i14jGiven v=3,6 and w=1,3 , a. Find projwv . (See Example 4) b. Find vectors v1 and v2 such that v1 , is parallel to w, v2 is orthogonal to w, and v1+v2=v . c. Using the results from part (b) show that v1 is parallel to w by finding a constant c such that v1=cw . d. Show that v2 is orthogonal to w. e. Show that v1+v2=v .Given v=4,3 and w=2,2 , a. Find projwv . b. Find vectors v1 and v2 such that v1 , is parallel to w, v2 is orthogonal to w, and v1+v2=v . c. Using the results from part (b) show that v1 is parallel to w by finding a constant c such that v1=cw . d. Show that v2 is orthogonal to w. e. Show that v1+v2=v .Given v=4i+9j and w=2ij , a. Find projwv . b. Find vectors v1 and v2 such that v1 , is parallel to w, v2 is orthogonal to w, and v1+v2=v .54PEA boat and trailer weighing a total of 450lb are parked on a boat ramp with an 18 angle of inclination. Assume that the weight of the boat and trailer is evenly distributed between two wheels. (See Example 5) a. Write the force vector F in terms of i and j representing the weight of the boat and trailer for a single tire. b. Find the component vector of F parallel to the ramp. Round values to 1 decimal place. c. Find the magnitude of the force needed to keep the trailer from moving down the ramp. Round to the nearest pound.A forklift is used to offload freight from a delivery truck. Together the forklift and its contents weigh800lb , and the weight is evenly distributed among four wheels .The ramp is inclined 14 from the horizontal. a. Write the force vector F in terms of i and j representing the weight against a single tire. b. Find the component vector of F parallel to the ramp. Round values to 1 decimal place. c. Find the magnitude of the force needed to keep the forklift from rolling down the ramp. Round to the nearest pound.57PE58PEa. Find the amount of work done in lifting a 50lb weight upward 5 ft b. If 200 N of force is applied in the direction of motion in moving an object 10 m, how much work is done?a. Find the amount of work done pulling an object horizontally 10ft using a force of 40lb in the direction of motion, b. If 2250N of force is applied in the direction of motion in moving an object50m , how much work is done?61PEFor Exercises 61-64, find the work done by the given force F acting in the direction from point A to point B. Assume that the units in the coordinate plane are in meters. A3,2,B5,4;F=250N63PE64PEFor Exercises 65-68, find the work w done by a force F in moving an object in a straight line given by the displacement vector D. (See Example 6) F=40i15jlb;D=30i+10jftFor Exercises 65-68, find the work w done by a force F in moving an object in a straight line given by the displacement vector D. (See Example 6) F=220i+350jlb;D=12i+16jftFor Exercises 65-68, find the work w done by a force F in moving an object in a straight line given by the displacement vector D. (See Example 6) F=26i+32jN;D=100i+120jm68PE69PEFor Exercises 69-72, find the work done by a force F (in lb) in moving an object in a straight line from point A to point B. Assume that the units in the coordinate plane are in feet. (See Example 6) A0,0,B5,4;F=2i+10j71PEFor Exercises 69-72, find the work done by a force F (in lb) in moving an object in a straight line from point A to point B. Assume that the units in the coordinate plane are in feet. (See Example 6) A14,20,B8,35;F=40i+26j73PE74PE75PEA child exerts a force of 28N on the handle of a small wagon. If the handle of the wagon is directed upwards 18 from the horizontal, find the amount of work done in moving the wagon 20m horizontally. Round to the nearest Nm .77PE78PEGiven v=22i55j and w=4i+10j , a. Find the angle between v and w. b. Are the vectors parallel? If yes, find a real number c such that v=cw .80PE81PEGiven c=5,12 and d=10,24 , a. Find the product cd . b. Find cd . c. Based on the results of parts (a) and (b), what do you know about the two vectors?If projwv=v , what do you know about v and w?If projwv=0 , what do you know about v and w?Given v=4i10j , a. Find two vectors parallel to v, one in the same direction as v and one in the opposite direction as v. Answers will vary. b. Find two vectors orthogonal to v. Answers will vary.Given v=3i9j , a. Find two vectors parallel to v, one in the same direction as v and one in the opposite direction as v. Answers will vary. b. Find two vectors orthogonal to v. Answers will vary.The dot product of vectors can be used in business applications. For Exercises 87-88, find the dot product and interpret the results. The components of n=500,330 represent the number of T-shirts and hats, respectively, in the inventory of a surf shop. The components of p=15,9 represent the price (in $) per T-shirt and hat, respectively. Find np and interpret the result.88PE89PE90PE91PEThe individual in Exercise 91 is directed to lose weight by his physician. After 6 months of exercise and dietary changes, his new weight is175lb . Recalculate parts (a) and (b) with the new body weight.93PEExplain what is meant by decomposing a nonzero vector v into orthogonal vectors.Prove that v-w2=v2+w22vw .98PE99PE100PEDetermine if the ordered pair is a solution to the system. 2x3y=05x+6y=1 a. 1,23 b. 6,4For Exercises 2-3, based on the slope-intercept form of the equations, determine the number of solutions to the system. y=35x4y=35x+1For Exercises 2-3, based on the slope-intercept form of the equations, determine the number of solutions to the system. y=2x+6y=12x6For Exercises 4-8, solve the system by using any method. If the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent. 4xy=72x+5y=19For Exercises 4-8, solve the system by using any method. If the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent. 5xy=192y0.2x+0.7y=1.7For Exercises 4-8, solve the system by using any method. If the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent. 9x2y=42x+4y=7For Exercises 4-8, solve the system by using any method. If the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent. 110x12y=12x=10y+6For Exercises 4-8, solve the system by using any method. If the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent. y=34x4yx=xShenika wants to monitor her daily calcium intake. One day she had 3 cups of milk and 1 cup of cooked spinach for a total of 1140 mg of calcium. The next day, she had 2 cups of milk and 112 cups of cooked spinach for a total of 960 mg of calcium. How much calcium is in 1 cup of milk and how much is in 1 cup of cooked spinach?How many liters of a 40 acid mixture and how many liters of a 10 acid mixture should be mixed to obtain 20L of a 22 acid mixture?A plane can travel 960mi in 2hr with a tail wind. The return trip against the wind takes 2hr and 40min. Find the speed of the plane in still air and the speed of the wind.A fishing boat captain charges $250 for an excursion. His fixed monthly expenses are $1200 for insurance, rent for the dock, and minor office expenses. He also has variable costs of $100 per excursion to cover gasoline, bait, and other equipment. a. Write a linear cost function representing the cost Cxin$ for the fishing boat captain to run x excursions per month. b. Write a linear revenue function representing the revenue Rxin$ for x excursions per month. c. Determine the number of excursions per month for the captain to break even. d. If 18 excursions are run in a given month, how much money will the fishing boat captain earn or lose?For Exercises 13-16, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. 3a4b+2c=172a+3b+c=14a+b3c=7For Exercises 13-16, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. 6x=245y14=7z3y4x3z=10For Exercises 13-16, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. x+2y+z=5x+yz=14x+7y+2z=16For Exercises 13-16, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. u+v+2w=12v5w=23u+5v+w=1Solve the system and write the general solution. 5x+2y+z=04x+3yz=06x+7y+z=018REEmily receives an inheritance of $20,000 and decides to invest the money. She puts some money in her savings account that earns 1.5 simple interest per year. The remaining money is invested in a bond fund that returns 4.5 and a stock fund that returns 6.2. She makes a total of $942 at the end of 1yr . If she invested twice as much in the bond fund as the stock fund, determine the amount that she invested in each fund.For Exercises 20-21, use a system of linear equations in three variables to find an equation of the form y=ax2+bx+c that defines the parabola through the points. 1,4,1,6,3,8For Exercises 20-21, use a system of linear equations in three variables to find an equation of the form y=ax2+bx+c that defines the parabola through the points. 1,2,2,1,3,10For Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 5x+22x2+8x+16For Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. x11x+2x1For Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 2x2+x10x3+5xFor Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 7x2+19x+152x3+3x2For Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 4x43x2+2x+5x2x+53x2+22For Exercises 22-27, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 2x3x2+8x16x4+5x2+4For Exercises 28-32, perform the partial fraction decomposition. x11x+2x1For Exercises 28-32, perform the partial fraction decomposition. 5x+22x2+8x+16For Exercises 28-32, perform the partial fraction decomposition. 2x4+7x3+13x2+19x+152x3+3x2For Exercises 28-32, perform the partial fraction decomposition. 2x2+x10x3+5xFor Exercises 28-32, perform the partial fraction decomposition. 2x3x2+8x16x4+5x2+4For Exercises 33-34, a. Graph the equations. b. Solve the system. yx2=1xy=3For Exercises 33-34, a. Graph the equations. b. Solve the system. y=x1x2+y2=5For Exercises 35-37, solve the system. 3x2y2=4x2+2y2=36For Exercises 35-37, solve the system. 2x2xy=24x2+3xy=9For Exercises 35-37, solve the system. y=8xy=x38REThe ratio of two numbers is 4 to 3. The sum of the squares of the numbers is 100. Find the numbers.The hypotenuse of a right triangle is 74ft and the sum of the lengths of the legs is 12ft. Find the lengths of the legs.A rectangular billboard has a perimeter of 72ft and an area of 288ft2. Find the dimensions of the billboard.Graph the solution set to the inequality. a. 3x+4y8 b. 3x+4y8Graph the solution set to the inequality. a. yx42 b. yx42For Exercises 44-48, graph the solution set. 5x+y8x+15For Exercises 44-48, graph the solution set. x3.5For Exercises 44-48, graph the solution set. 32y+14For Exercises 44-48, graph the solution set. x2+y+224For Exercises 44-48, graph the solution set. y2Determine if the given ordered pair is a solution to the system of inequalities. x+2y43x4y6 a. 0,1 b. 1,4For Exercises 50-53, graph the solution set. If there is no solution, indicate that the solution set is the empty set. y12x+13x+2y4For Exercises 50-53, graph the solution set. If there is no solution, indicate that the solution set is the empty set. x2+y29x12+y24For Exercises 50-53, graph the solution set. If there is no solution, indicate that the solution set is the empty set. yx23y1x+y3For Exercises 50-53, graph the solution set. If there is no solution, indicate that the solution set is the empty set. 3exyx21Let x represent the number of hours that Gordon spends tutoring math, and let y represent the number of hours that he spends tutoring English. For parts (a)-(d), write an inequality to represent the given statement. a. Gordon has at most 12hr to tutor per week. b. The amount of time that Gordon spends tutoring English is at least twice the amount of time he spends tutoring math. c. The number of hours spent tutoring math cannot be negative. d. The number of hours spent tutoring English cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).At a home store, one sheet of 38-in. sanded pine plywood costs $24. One sheet of 14-in. sanded pine plywood costs $20. Write an objective function z=fx,y that represents the total cost for x38-in. sheets and y14-in. sheets.For the feasible region given in the figure and the objective function z=36x+50y, a. Determine the values of x and y that produce the maximum value of the objective function. b. Determine the maximum value of the objective function.For the given constraints and the objective function, z=55x+40y, a. Graph the feasible region and identify the vertices. x0,y02x+y185x+4y60 b. Determine the values of x and y that produce the minimum value of the objective function on the feasible region. c. Determine the minimum value of the objective function on the feasible region.A fitness instructor wants to mix two brands of protein powder to form a blend that limits the amount of fat and carbohydrate but maximizes the amount of fiber. The nutritional information is given in the table for a single scoop of protein powder. Suppose that the fitness instructor wants to make at most 180 scoops of the mixture. She also wants to limit the amount of fat to 480g and she wants to limit the amount of carbohydrate to 696g. a. Determine the number of scoops of each type of powder that will maximum the amount of fiber. b. What is the maximum amount of fiber? c. If the fiber content were reversed between the two brands (that is, 8g for brand A and 10g for brand B ), then how much of each type of protein powder should be used to maximize the amount of fiber?For Exercises 1-3, determine if the ordered pair or ordered triple is a solution to the system. x5y=3y=2x12 a. 7,2 b. 3,0For Exercises 1-3, determine if the ordered pair or ordered triple is a solution to the system. 2x3y+z=55x+y3z=18x+2y+5z=8 a. 0,1,2 b. 3,0,1For Exercises 1-3, determine if the ordered pair or ordered triple is a solution to the system. 2x4y93x+y4 a. 6,1 b. 1,4For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. x=54y3x+7y=4For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 0.2x=0.35y2.50.16x+0.5y=5.8For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. x25y=3105x=2y+32For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 7xy=35y43xy=2xFor Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. a+6b+3c=142a+b2c=83a+2b+x=8For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. x+4z=103y2z=92x+5y=21For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 2xy+z=3x3y=2x+2y+z=7For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. x42+y2=25xy=3For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 5x2+y2=14x22y2=17For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 2xyy2=243xy+2y2=38For Exercises 4-14, solve the system. If the system does not have one unique solution, also state whether the system is inconsistent or whether the equations are dependent. 1x+32y1=73x+3+1y1=7Solve the system and write the general solution. x2z=6y+3z=2x+y+z=8At a candy and nut shop, the manager wants to make a nut mixture that is 56 peanuts. How many pounds of peanuts must be added to an existing mixture of 45 peanuts to make 20lb of a mixture that is 56 peanuts?Two runners begin at the same point on a 400-m track. If they run in opposite directions they pass each other in 40sec . If they run in the same direction, they will meet again in 200sec. Find the speed of each runner.Dylan invests $15,000 in three different stocks. One stock is very risky and after 1yr loses 8. The second stock returns 3.2, and a third stock returns 5.8. At the end of 1yr, the total return is $274. If he invested $2000 more in the second stock than in the third stock, determine the amount he invested in each stock.The difference of two positive numbers is 3 and the difference of their squares is 33. Find the numbers.A rectangular television screen has a perimeter of 154in. and an area of 1452in2. Find the dimensions of the screen.Use a system of linear equations in three variables to find an equation of the form y=ax2+bx+c that defines the parabola through the points 1,1,2,1, and 1,7.For Exercises 22-23, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 15x+153x2+x2For Exercises 22-23, set up the form for the partial fraction decomposition. Do not solve for A,B,C , and so on. 5x6+3x54x3+x3x3x3x2+5x+12For Exercises 24-28, perform the partial fraction decomposition. 12x292x2+11x+15For Exercises 24-28, perform the partial fraction decomposition. 6x+8x2+4x+4For Exercises 24-28, perform the partial fraction decomposition. x46x3+4x2+20x32x34x2For Exercises 24-28, perform the partial fraction decomposition. x22x21x3+7xFor Exercises 24-28, perform the partial fraction decomposition. 7x3+4x2+63x+15x4+11x2+18For Exercises 29-33, graph the solution set. 2x+y6yFor Exercises 29-33, graph the solution set. x+32+y29For Exercises 29-33, graph the solution set. x4For Exercises 29-33, graph the solution set. x+y42xy2For Exercises 29-33, graph the solution set. yx2+5y1x+y3A donut shop makes a profit of $2.40 on a dozen donuts and $0.55 per muffin. Write an objective function z=fx,y that represents the total profit for selling x dozen donuts and y muffins.For the feasible region given and the objective function z=4x+5y, a. Determine the values of x and y that produce the minimum value of the objective function on the feasible region. b. Determine the minimum value of the objective function on the feasible region.For the given constraints and objective function, z=600x+850y, a. Graph the feasible region and identify the vertices. x0,y0x+y48y3x b. Determine the values of x and y that produce the maximum value of the objective function on the feasible region. c. Determine the maximum value of the objective function on the feasible region.A weight lifter wants to mix two types of protein powder. One is a whey protein and one is a soy protein. The fat carbohydrate, and protein content (in grams) for 1 scoop of each powder is given in the table. Suppose that the weight lifter wants to make at most 60 scoops of a protein powder mixture. Furthermore, he wants to limit the total fat content to at most 150g and the total carbohydrate content to at most 216g. a. Determine the number of scoops of each type of powder that will maximize the total protein content under these constraints. b. What is the maximum total protein content? c. If the protein content were reversed between the two brands (that is, 18g for the whey protein and 20g for the soy protein), then how much of each type of protein powder should be used to maximize the amount of protein?For Exercises 1-5, solve the equation. 2xx+2=5x+7For Exercises 1-5, solve the equation. 2t+8t=4For Exercises 1-5, solve the equation. x2427x2460=04CREFor Exercises 1-5, solve the equation. 50e2x+1=2000For Exercises 6-7, solve the inequality. Write the solution set in interval notation. x+4x21For Exercises 6-7, solve the inequality. Write the solution set in interval notation. 3x+218Find the partial fraction decomposition, 5x+17x26x+9Given fx=2x23x and gx=5x+1, a. Find fgx. b. Find gfx.Given fx=x23, write an equation for f1x.Use a calculator to approximate the value of log5256. Round to 4 decimal places.Given fx=12x3+4x2+2, find the average rate of change on the interval 1,3.Write an equation of the line perpendicular to the line x+3y=6 and passing through the point 2,1.Find all zeros of fx=x42x3+10x218x+9 and state the multiplicity of each zero.For Exercises 15-16, a. Write the domain in interval notation. b. Write the range in interval notation. fx=2x13For Exercises 15-16, a. Write the domain in interval notation. b. Write the range in interval notation. fx=lnx3For Exercises 17-19, solve the system. 3x=5y+1y=35x+4For Exercises 17-19, solve the system. 5a+2b+3c=103a+b2c=7a+4b4c=3For Exercises 17-19, solve the system. 2x2+3y2=105x2+2y2=13Given fx=x2+5x+1, find the difference quotient.Shen invested $8000. After 5yr with interest compounded continuously, the account is worth $10,907.40. a. Write a model of the form At=Pert, where At represents the amount (in $ ) in the account if P dollars in principal is invested at interest rate r for t years. b. How long will it take for the investment to double? Round to the nearest tenth of a year.The variable y varies jointly as x and the square of z. If y is 36 when x is 10 and z is 3, find the value of y when x=12 and z is 4.Determine if the ordered pair is a solution of the system. 3xy=10x+14y=1 a. 2,4 b. 13,9Solve the system by using the substitution method. 3x+4y=5x3y=63SPSolve the system by using the addition method. 2x2y=y+1412x+76y=1335SP6SPHow many ounces of 20 and 35 acid solution should be mixed to produce 15 oz of 30 acid solution?A boat takes 3hr to go 24mi upstream against the current. It can go downstream with the current a distance of 48mi in the same amount of time. Determine the speed of the boat in still water and the speed of the current.9SPTwo or more linear equations taken together make up a of linear equations.A to a system of equations in two variables is an ordered pair that is a solution to each individual equation in the system.Two algebraic methods to solve a system of linear equations in two variables are the method and the method.