9.1E: Exercises
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Exercises
(Review Exercises) In Exercises 1 - 8, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
- {x+2y=5x=6
- {2y−3x=1y=−3
- {x+2y4=−53x−y2=1
- {23x−15y=312x+34y=1
- {12x−13y=−12y−3x=6
- {x+4y=6112x+13y=12
- {3y−32x=−15212x−y=32
- {56x+53y=−73−103x−203y=10
In Exercises 9 - 26, put each system of linear equations into triangular form and solve the system if possible. Classify each system as consistent independent, consistent dependent, or inconsistent.
- {−5x+y=17x+y=5
- {x+y+z=32x−y+z=0−3x+5y+7z=7
- {4x−y+z=52y+6z=30x+z=5
- {4x−y+z=52y+6z=30x+z=6
- {x+y+z=−17y−3z=0
- {x−2y+3z=7−3x+y+2z=−52x+2y+z=3
- {3x−2y+z=−5x+3y−z=12x+y+2z=0
- {2x−y+z=−14x+3y+5z=15y+3z=4
- {x−y+z=−4−3x+2y+4z=−5x−5y+2z=−18
- {2x−4y+z=−7x−2y+2z=−2−x+4y−2z=3
- {2x−y+z=12x+2y−z=13x+6y+4z=9
- {x−3y−4z=33x+4y−z=132x−19y−19z=2
- {x+y+z=42x−4y−z=−1x−y=2
- {x−y+z=83x+3y−9z=−67x−2y+5z=39
- {2x−3y+z=−14x−4y+4z=−136x−5y+7z=−25
- {2x1+x2−12x3−x4=16−x1+x2+12x3−4x4=−53x1+2x2−16x3−3x4=25x1+2x2−5x4=11
- {x1−x3=−22x2−x4=0x1−2x2+x3=0−x3+x4=1
- {x1−x2−5x3+3x4=−1x1+x2+5x3−3x4=0x2+5x3−3x4=1x1−2x2−10x3+6x4=−1
- Find two other forms of the parametric solution to Exercise 11 above by reorganizing the equations so that x or y can be the free variable.
- A local buffet charges $7.50 per person for the basic buffet and $9.25 for the deluxe buffet (which includes crab legs.) If 27 diners went out to eat and the total bill was $227.00 before taxes, how many chose the basic buffet and how many chose the deluxe buffet?
- At The Old Home Fill’er Up and Keep on a-Truckin’ Cafe, Mavis mixes two different types of coffee beans to produce a house blend. The first type costs $3 per pound and the second costs $8 per pound. How much of each type does Mavis use to make 50 pounds of a blend which costs $6 per pound?
- Skippy has a total of $10,000 to split between two investments. One account offers 3% simple interest, and the other account offers 8% simple interest. For tax reasons, he can only earn $500 in interest the entire year. How much money should Skippy invest in each account to earn $500 in interest for the year?
- A 10% salt solution is to be mixed with pure water to produce 75 gallons of a 3% salt solution. How much of each are needed?
- At The Crispy Critter’s Head Shop and Patchouli Emporium along with their dried up weeds, sunflower seeds and astrological postcards they sell an herbal tea blend. By weight, Type I herbal tea is 30% peppermint, 40% rose hips and 30% chamomile, Type II has percents 40%, 20% and 40%, respectively, and Type III has percents 35%, 30% and 35%, respectively. How much of each Type of tea is needed to make 2 pounds of a new blend of tea that is equal parts peppermint, rose hips and chamomile?
- Discuss with your classmates how you would approach Exercise 32 above if they needed to use up a pound of Type I tea to make room on the shelf for a new canister.
- If you were to try to make 100 mL of a 60% acid solution using stock solutions at 20% and 40%, respectively, what would the triangular form of the resulting system look like? Explain.
Answers
-
Consistent independent
Solution (6,−12) -
Consistent independent
Solution (−73,−3) -
Consistent independent
Solution (−167,−627) -
Consistent independent
Solution (4912,−2518) -
Consistent dependent
Solution (t,32t+3)
for all real numbers t -
Consistent dependent
Solution (6−4t,t)
for all real numbers t -
Inconsistent
No solution -
Inconsistent
No solution
Because triangular form is not unique, we give only one possible answer to that part of the question. Yours may be different and still be correct.
- {x+y=5y=7
Consistent independent
Solution (−2,7)
- {x−53y−73z=−73y+54z=2z=0
Consistent independent
Solution (1,2,0)
- {x−14y+14z=54y+3z=150=0
Consistent dependent
Solution (−t+5,−3t+15,t)
for all real numbers t
- {x−14y+14z=54y+3z=150=1
Inconsistent
No solution
- {x+y+z=−17y−3z=0
Consistent dependent
Solution (−4t−17,3t,t)
for all real numbers t
- {x−2y+3z=7y−115z=−165z=1
Consistent independent
Solution (2,−1,1)
- {x+y+2z=0y−32z=6z=−2
Consistent independent
Solution (1,3,−2)
- {x−12y+12z=−12y+35z=350=1
Inconsistent
no solution
- {x−y+z=−4y−7z=17z=−2
Consistent independent
Solution (1,3,−2)
- {x−2y+2z=−2y=12z=1
Consistent independent
Solution (−3,12,1) - {x−12y+12z=12y−23z=0z=1
Consistent independent
Solution (13,23,1)
- {x−3y−4z=3y+1113z=4130=0
Consistent dependent
Solution (1913t+5113,−1113t+413,t)
for all real numbers t
- {x+y+z=4y+12z=320=1
Inconsistent
no solution
- {x−y+z=8y−2z=−5z=1
Consistent independent
Solution (4,−3,1)
- {x−32y+12z=−12y+z=−1120=0
Consistent dependent
Solution (−2t−354,−t−112,t)
for all real numbers \(t\
- {x1+23x2−163x3−x4=253x2+4x3−3x4=20=00=0
Consistent dependent
Solution (8s−t+7,−4s+3t+2,s,t)
for all real numbers s and t
- {x1−x3=−2x2−12x4=0x3−12x4=1x4=4
Consistent independent
Solution (1,2,3,4)
- {x1−x2−5x3+3x4=−1x2+5x3−3x4=120=10=0
Inconsistent
No solution
- If x is the free variable then the solution is (t,3t,−t+5) and if y is the free variable then the solution is (13t,t,−13t+5).
- 13 chose the basic buffet and 14 chose the deluxe buffet.
- Mavis needs 20 pounds of $3 per pound coffee and 30 pounds of $8 per pound coffee.
- Skippy needs to invest $6000 in the 3% account and $4000 in the 8% account.
- 22.5 gallons of the 10% solution and 52.5 gallons of pure water.
- 43−12t pounds of Type I, 23−12t pounds of Type II and t pounds of Type III where 0≤t≤43.