Systems of Equaitons Practice Quiz

Some questions (c) 2017 by TEKS Resource System.

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1 Which table shows a solution set to the systems of equations shown?

y = x2 + 1

2x2 – 2y = − 2

A x y− 2 50 12 54 17x y− 2 − 50 − 12 − 54 − 17

B x y− 2 50 12 54 17x y− 2 50 12 54 17

C x y− 2 40 02 44 16x y− 2 − 50 − 12 − 54 − 17

D x y− 2 40 02 44 16

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x y− 2 50 12 54 17

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2 Simone has discovered a blow-out sale at a shop in the mall. The cost of 2 sweaters,5 shirts, and 3 pairs of pants is $172. The cost of 5 sweaters and 8 pairs of pants is$269. The cost of 4 shirts and 1 pair of pants is $74. If x represents sweaters, yrepresents shirts, and z represents pants, which systems of equations can be used todetermine the sale price of the sweaters, the shirts, and the pants?

F 2x + 5y + 3z = 1725x + 8z = 2694y + z = 74

G 2x + 5y + 3z = 1725x + y + 8z = 269x + 4y + z = 74

H x + y + z = 172x + z = 269y + z = 74

J 2x + 5y + 3z = 1725x – 8z = 2694y – z = 74

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3 The sum of three numbers is 27. The product of the first number times the thirdnumber is 56. The quotient of the second number divided by the first number is 1.5. Ifthe first number is represented by x, the second number is represented by y, and thethird number is represented by z, which of the systems of equations below can beused to determine the three numbers?

A x + y + z = 27xy = 56x – 1.5y = 0

B x + y + z = 27xz = 56–1.5x + y = 0

C x + y + z = 27xz = 56

= 1.5

D x + y + z = 27xz = 56

= y

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4 At Finky Feed Store, Gerby purchased 7 pounds of cracked corn, 3 pounds ofsunflower seed, and 8 pounds of mixed bird seed for $31.00. Michael purchased 5pounds of cracked corn, 4 pounds of sunflower seed, and 10 pounds of mixed birdseed for $32.75. Jayleen purchased 4 pounds of sunflower seed and 12 pounds ofmixed bird seed for $27.00. If c represents cracked corn, s represents sunflowerseeds, and b represents bird seed, which of the systems of equations below can beused to determine the price per pound of cracked corn, sunflower seed, and bird seed?

F c + s + b = 18(31)c + s + b = 19(32.75)s + b = 16(27)

G 7c + 3s + 8b = 315c + 4s + 10b = 32.754c + 12s = 27

H 16(c + s + b) = 3119(c + s + b) = 32.7516(s + b) = 27

J 7c + 3s + 8b = 315c + 4s + 10b = 32.754s + 12b = 27

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5 The sum of three numbers is 19. Two times the first number increased by six timesthe second number plus the third number is 12. The sum of three times the firstnumber and eight times the second number, increased by the third number is –14. Ifthe first number is represented by x, the second number is represented by y, and thethird number is represented by z, which of the systems of equations below can beused to determine the three numbers?

A x + y + z = 192x • 6y + z = 123x + 8y • z = –14

B x + y + z = 192x + 6y = z + 123x + 8y = z – 14

C x + y + z = 192x + 6y + z = 123x + 8y + z = –14

D x + y + z = 19x + 2y + 6z = 12x + 3y + 8z = –14

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6 Dot’s Tea Room has three items on its lunch special: sandwiches, salads, and personalpizzas. All sandwiches are the same price; all salads are the same price; and allpersonal pizzas are the same price. On Monday the office ordered 5 sandwiches, 7salads, and 3 personal pizzas for $116.00. On Wednesday the office ordered 3sandwiches, 10 salads, and 2 personal pizzas for $114.75. On Friday the officeordered 6 sandwiches, 12 salads, and 5 personal pizzas for $177.75.

Part AIf w represents the cost of each sandwich, s represents the cost of each salad, and prepresents the cost of each pizza, formulate a system of three equations in threevariables to represent the total cost each day.

Part BSolve the system of three equations in three variables to determine the cost of eachlunch item using an appropriate solution method.

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Part CJustify the solution to the system of three equations in three variables in terms of theproblem situation.

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7 Suzette solved a system of three linear equations in three variables using substitution.The steps she used to solve the system are shown below.

3x + 5y + 2z = 32x + 3z = 2x – y = –7

Step 1x = y – 72(y – 7) + 3z = 23(y – 7) + 5y + 2z = 3

Step 22y + 3z = 164y + z = 12

Step 32y + 3z = 16–12y – 3z = –36

Step 4y = 2x = (2) – 7x = –52(–5) + 3z = 2–10 + 3z = 23z = 12z = 4Therefore:x = –5, y = 2, z = 4

Did Suzette make a mistake in solving the system of three linear equations in threevariables, and if so, in which step did the mistake occur?

A Suzette made a mistake in Step 1.

B Suzette made a mistake in Step 2.

C Suzette made a mistake in Step 3.

D Suzette did not make a mistake in solving the system of three linear equations inthree variables.

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Part CUse the model function that best represents the problem situation to predict theapproximate number of hours it will take the amount of antibiotic in Tyrone’s systemto drop to 100 mg.

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Part DAccording to the model function that best represents the problem situation, will theamount of antibiotic in Tyrone’s system ever drop to 0 mg. Explain your response.

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9 The Spanish Club is selling plates of food as a fundraiser. A plate with 2 tacos, 2cheese enchiladas, and 1 tamale costs $8.50. A plate with 3 tacos, 2 cheeseenchiladas, and 2 tamales costs $13.00. A plate with 1 taco, 3 cheese enchiladas, and2 tamales costs $11.25.

Part AIf x represents the cost of a taco, y represents the cost of an enchilada, and zrepresents the cost of a tamale, formulate a system of three equations in threevariables to determine the cost for each taco, enchilada, and tamale.

Part BSolve the system of three equations in three variables to determine how much theSpanish Club is charging for each taco, enchilada, and tamale on the plates of food.

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Part CJustify the solution to the system of three equations in three variables in terms of theproblem situation.

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10 Simone went to the outlet mall and found a blow-out sale. Sweaters were $5 andshirts were $3. The product of the number of sweaters and shirts she purchased is30. She paid $45 for her sweaters and shirts.

A. Ignoring tax, set up the system of equations that can be used to find the numberof sweaters, and shirts Simone bought.

B. Solve the system.

11 Mandy bought a desktop computer system, to start her business from home, for$4,995. It is expected to depreciate at a rate of 11.5% per year. What will be thevalue of the computer system in 3 years? Round your answer to the nearest dollar.

A $4,421

B $3,462

C $2,812

D $3,802

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12 The percent concentration of poisonous gas in a room is at 45%. An exhaust systemis being used to clear the poisonous gas. The exhaust system fans are decreasingthe concentration exponentially by 5% each hour. The area cannot be entered whenthe concentration is above 15%.

Part A

Determine an inequality that could be used to express the number of hours, t, forwhich the room is safe to enter.

Part B

Solve the inequality on the graph below.

Part C

Verify the solution using a table.

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13 Re-write the second equation in thefollowing system to allow thesystem to be solved by substitution.

2x + 5y=17 x – 5y=15

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15 The system of three linear equations in three variables below is to be solved usingtechnology with matrices.

2x + z = 5y – 17.754x – 3y = 5z + 8.258x + 2z = y + 2z + 6.5

Which matrices could be used to solve the system of equations?

A

B

C

D

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16 Solve the system of three linear equations in three variables shown below using anappropriate method.

5z = 2y4y = 2 – x3x + 2y = 5z + 12

Which set of points, (x, y, z), represents the solution to the system of three linearequations in three variables?

F (8, 2.5, 1)

G

H

J

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17 The graph of the system of two linear inequalities in two variables is shown below.

Which system of inequalities can be used to represent the problem?

A 2x – 3y > 62x + y ≤ 2

B 2x – 3y < 62x + y ≤ 2

C 3x – 2y < 6x + 2y ≤ 2

D 2x – 3y > 32x + y ≤ 2

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18 A system of two linear inequalities in two variables is shown below.

2y ≥ x – 24x – 3y > 6

Which graph represents the region of solutions to the system of inequalities?

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F

G

H

J

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19 A system of three linear inequalities in two variables is shown below.

3y + 5 ≤ 207x – 2y > 123x + 5y ≥ 25

Which graph represents the correct region of solutions to the system of inequalities?

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A

B

C

D

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21 Randi is sky diving with her parachute open and is falling at a constant rate of4.8 ft/sec. When Randi is 40.6 feet above ground, her friend, standing on a rooftop15 feet high, tosses her a pillow to cushion her landing. The height of the pillow as afunction of elapsed time is represented by the function below.

h(t) = –16t2 + 40t + 15

Part AWhat linear function can be used to find the parachutist’s height in terms of theelapsed time since the friend threw the pillow?

Part BSolve the system of equations algebraically. Round answers to the nearesthundredth, if necessary.

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Part CSolve the system of equations graphically. Round answers to the nearest hundredth,if necessary.

Part DWhich method of solving best represents the meaning and reasonableness of thesolution in terms of the problem situation? Justify your reasoning.

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22 Anita algebraically solved the system of a linear equation and a quadratic equation intwo variables shown below.

x2 + 5 = y + 6x2x = y – 5

Anita’s solution was (0, 5) and (8, 21). Which method can Anita use to determinethe reasonableness of her solution to the system of equations?

F Anita can solve each equation for y, graph both equations on a coordinate plane,and determine the intersection points.

G Anita can solve each equation for y, create a table of values for each equation,and determine the points in the table where the equations have the same yvalue.

H Anita can substitute the point (0, 5) into both equations and confirm that it yieldsa true equation, and also substitute the point (8, 21) into both equations andconfirm that it yields a true equation.

J All of the above

23 Mandy bought a desktop computer system to start her business from home for$4,995. It is expected to depreciate at a rate of 10% per year. After how manyyears will the value of her home computer system depreciate to approximately$2,150?

A 32 years

B 5 years

C 9 years

D 8 years

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25 A system of three linear inequalities in two variables is shown below.

5y + 3x ≤ 152x – 7y > 145x + 2y ≥ 4

Which graph represents the correct region of solutions to the system of inequalities?

A

B

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C

D

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27 Solve the following system ofequations by an appropriatemethod.

xy = 8y = x – 2

A (4, 2) only

B (-4, -2) and (2, 4)

C (4, 2) and (-2, -4)

D no solution

28 The area of a rectangle is 348square feet. The length is 5 feetlonger than twice the width.Which system of equations can besolved to find the length (L) and thewidth (W) of the rectangle?

F 2L + 2W = 348W = 2L + 5

G (L)(W) = 348W = 2L + 5

H (L)(W) = 348L = 2W + 5

J 2L + 2W = 348L = 2W + 5

26 Solve the system of three equations in three variables by an appropriate method.

5x + 2z = 3y – 243x = 5 + 4z15 – 10z = 7y

Which three-dimensional point in the form (x, y, z) could be represented by thesolution of system of three equations in three variables?

F (–13.04, –6.78, 18.26)

G (–1, 5, –2)

H (–4.11, –4.04, 4.33)

J (–2.32, 02.13, –2.99)

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29 A cardboard box has a base withdimensions of x and y. Set up asystem of equations in two variableswhich represents the situationdescribed. The x-side of the base is4 inches less than three times theother side. The area is 120 squareinches. Do not solve.

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30 Which system of equations describes the graph?

F y = x2 − 3x + 22x + y = 3

G y = x2 + 3x + 22x + y = 3

H y = x2 − 3x + 22x − y = − 3

J y = x2 + 3x + 22x − y = − 3

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31 Which of the following describes the solution set to the system shown below?

y = x2

y = x + 2

I. Finite and consistentII. Infinite and inconsistentIII. DependentIV. Independent

A I only

B II only

C II and III

D I and IV

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32 The perimeter of a rectangle is 28 inches. The area of the rectangle is 40 squareinches. Let l and w represent the dimensions of the rectangle.A. Set up a system of equations that could be used to find l and w.

B. Find the dimensions of the rectangle.

33 Solve the system of equations and describe the solution.

3x + 5y = 106x + 10y = 15

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34 Beth proposed a business plan, for an advertising company, to a bank that providesqualified applicants with small business loans. Included in her plan is the sale ofmarketing mouse pads. The following graph illustrates the break-even analysisprojected for the monthly sales of the mouse pads after fixed and total costs.

Given the system of equations for total costs and sales, how much money in salesmust Beth earn to break-even?

F $25,000

G $20,000

H $19,500

J $5,000

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36 If (-4, 5) is the only solution to asystem of two linear equations, thenthe graphical solution would show:

F 2 intersecting lines

G at least one vertical line

H 2 identical lines

J 2 parallel lines

37 If a system of two linear equationshad no solutions, the graph wouldshow which of the following?

A 2 intersecting lines

B at least one vertical line

C 2 identical lines

D 2 parallel lines

38 Which system of inequalitiesdescribes the graph?

F y > 2xx + y ≥ 3

G y ≤ 2xy > x + 3

H y < 2xx + y ≥ 3

J y ≤ 2xx + y > 3

35 The table below illustrates highlights of her business plan, including projected sales, s,and net profit, p.

Year (n) Sales (s) Net Profit (p)1 $200,000 $25,0002 $350,000 $75,0003 $500,000 $125,000

Write a system of linear equations in function notation that can be used to modelprojected sales, s(n), and net profit, p(n).

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40 If you are solving a system for thenumber of Class A and Class Btickets sold on a charter flight, whattypes of numbers would NOT beconsidered reasonable and why?

41 Set up a system of equations thatcould be used to solve the followingscenario. Do not actually solve theproblem.A total of 698 tickets were sold foran event. Three times as many goldtickets, g, were sold as silver tickets,s. The total number of gold andsilver tickets sold was twice thenumber of bronze tickets, b, sold.

42 The first of two numbers minus fourtimes a second number is at least -4. Three times the first number plusthe second number is more than -3.Using x as the first and y as thesecond number, set up a system ofinequalities and locate all possiblesolutions for x and y graphically.

39 The school jazz ensemble collected $1,670 for 300 tickets sold for their concert.Adult tickets were $9 each and student tickets were $4 each. Write a system ofequations that could be used to solve this problem.

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43 Solve the system of equationsbelow by using an appropriatemethod.

2x + 3y = 34x - 9y = -46

44 Solve the following system ofequations by an appropriatemethod.

5x - 2y = -39-2x + y = 18

45 Rewrite the first equation in thefollowing system so that you coulduse the Elimination Method ofsolving:

x – 2y = 123x + 4y = 1

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46 A cardboard box has a base with dimensions as shown in the figure below.

The area of the bottom of the cardboard box is 20 square inches. The lateral surfacearea of the cardboard box is 108 square inches. The volume of the cardboard box is120 cubic inches. Formulate a system of three equations in three variables that canbe used to determine the lengths of the three sides of the cardboard box.

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47 Susie’s Sweet Shop sells chocolate boxes that contain three types of chocolatetruffles: solid chocolate truffles, cream center chocolate truffles, and chocolatetruffles with nuts. A box containing 5 of each type of truffle costs $41.25. A boxcontaining 10 solid chocolate truffles, 5 cream center chocolate truffles, and 10chocolate truffles with nuts costs $68.75. A box with only 24 truffles evenly dividedbetween solid chocolate truffles and chocolate truffles with nuts costs $66.00. If srepresents the cost per solid chocolate truffle, c represents the cost per creamcenter chocolate truffle, and n represents the cost per chocolate truffle with nuts,which system of equations can be used to determine the cost of each type ofchocolate truffle?

A 5s + 5c + 5n = 5(41.25)10s + 5c + 10n = 25(68.75)12s + 12n = 24(66)

B 5s + 5c + 5n = 41.2510s + 5c + 10n = 68.758s + 8n + 8n = 66

C 5s + 5c + 5n = 41.2510s + 5c + 10n = 68.7512s + 12n = 66

D 5s – 5c – 5n = 41.2510s – 5c – 10n = 68.758s – 8n – 8n = 66

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48 Miss Croft made snack bags for the picnic that contain three types of snacks:packages of crackers, packages of cookies, and candy bars. A snack bag containing6 of each type of snack costs $21.00. A snack bag containing 8 packages ofcrackers, 5 packages of cookies, and 10 candy bars costs $26.00. A snack bagcontaining 5 packages of crackers, 4 packages of cookies, and 7 candy bars costs$18.50. Solve a system of three linear equations in three variables to determine thecost of one candy bar.

Record your answer and fill in the bubbles on your answer document. Be sure to usethe correct place value.

49 April is going to the bakery to purchase donuts and cinnamon rolls for the officepersonnel. She must purchase at least 30 items. She has collected only $30, so totalcost must be no more than $30. If d represents donuts that cost $0.75 each and rrepresents cinnamon rolls that cost $1.25 each, which system of inequalities can beused to represent the region for the number of donuts and cinnamon rolls April couldpurchase?

A d + r ≤ 300.75d + 1.25r ≥ 30

B d + r ≥ 300.75d + 1.25r ≤ 30

C d + r ≥ 301.25d + 0.75r ≤ 30

D d + r ≤ 301.25d + 0.75r ≥ 30

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50 The first of two numbers minus four times a second number is at least –4. Threetimes the first number plus the second number is more than –3. If x represents thefirst number and y represents the second number, which system of inequalities bestrepresents the problem situation?

F 2x – 4y ≥ –43x + 2y > –3

G x – 4y ≤ –43x + y > –3

H 2x – 4y < –43x + y > –3

J x – 4y ≥ –43x + y > –3

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51 Janet is making brownies and chocolate chip cookies for the band fund raiser. Eachbrownie recipe makes 18 brownies. Each cookie recipe makes 36 cookies. She hastold the organizers of the bake sale she will make at least 288 brownies and

chocolate chip cookies. Each brownie recipe requires cups of flour. Each chocolate

chip cookie recipe requires cups of flour. Janet has 6 four-pound sacks of flour

available to make cookies and brownies. Each pound of flour yields cups of flour.If b represents the number of brownie recipes Janet can make and c represents thenumber of chocolate chip cookie recipes Janet can make, which system ofinequalities can be used to represent the region for the number of brownies andcookies Janet can make for the band fund raiser?

Ac ≤ – b +

c ≥ – b +8

B 18b + 36c ≥ 288

b + c ≤ 24

C 18b + 36c ≥ 288

b + c ≤ 80

D b + 2c ≥ 165b + 6c ≤ 40

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52 The Middletown High School Booster Club is having a fund raiser to pay for theconstruction of a sculpture to place in the entry of the school. For the fundraiser theywill be selling bags of fruit and boxes of honey that contain three types of honey. Theboosters will make $15 on each bag of fruit and $7.50 on each box of honey. Due tolack of storage space at the school, no more than 100 items can be purchased forsale. The Middletown High School Booster Club hopes to make at least $1,200 onthe fund raiser.

Part ALet x represent the number of bags of fruit and y represent the number of boxes ofhoney. Write a system of inequalities to represent the constraints of the situation.

Part BDraw a graph to represent the feasible region that meets all the constraints.

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53 During the summer, Tuscany Pies makes peach pies. The peaches used at TuscanyPies come from Ritz Peach Farms and Ormond’s Orchards. Tuscany Pies is analyzingdata to determine the most economical method for purchasing and processing thepeaches for making the pies. The following conditions impact the economics ofproducing peach pies at Tuscany Pies.

Tuscany Pies must process at least 500 pounds of peaches each week to meetdemands.Tuscany Pies has limited labor and storage and can process a maximum of 750pounds a week.Each shipment of peaches contains a certain amount of waste product (such as skinand pits) that will not be used. Tuscany Pies wants no more than 225 pounds per

week of such waste. Tuscany Pies estimates that of every pound of peaches from

Ritz Peach Farms is waste, and of every pound of peaches from Ormond’sOrchards is waste.

Part ALet x represent the pounds of peaches purchased from Ritz Peach Farms in a weekand y represent the number of pounds of peaches purchased from Ormond’sOrchards in a week. Write a system of inequalities to represent the constraints of thesituation.

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Part BDraw a graph to represent the feasible region that meets all the constraints.

Part CRitz Peach Farms charges $3.75 per pound for peaches, and Ormond’s Orchardscharges $3.50 per pound for peaches. What are the maximum and minimum coststhat Tuscany Pies should budget each week for peaches? How might the minimumand maximum impact their business? Justify your reasoning.

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54 Mason, standing just outside his apartment building, throws a baseball upward froman initial height of 5 feet above ground level at an initial velocity of 90 ft/sec. Theheight of the baseball in terms of elapsed time is a quadratic relationship. The

generalized equation is f(x) = –16x2 + v0x + h0, where v0 represents initial velocity

and h0 represents initial height. Bill, on his third floor balcony 40 feet above ground

level, launches his drone helicopter that ascends at a constant rate of 25 ft/sec. IfMason throws the baseball at the same time that Bill begins the ascent flight with hishelicopter, formulate a system of equations to determine at what elapsed time(s)and height(s) the baseball would possibly strike the helicopter.

F yb = –16x2 + 5x + 90

yh = 25x + 40

G yb = –16x2 + 90x + 5

yh = 25x + 40

H

yb = –16x2 + 90x + 5

yh = 25x2 + 40

J yb = –16x2 + 5x + 90

yh = 25x2 + 40

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55 In the year 2000, Kevin and Keith each received $5,000. Kevin decided to put his$5,000 in his house safe and each year add $500 to his money stash. Keith decidedto invest his $5,000 in computer stock. The value of Keith’s stock climbed accordingto a parabolic curve until it peaked at $17,500 in 2010. By 2014, the stocks hadfallen to a value of $15,500. Formulate a system of equations to determine thenumber of years it would take for the amount Kevin has in his money stash to beequal to the value of Keith’s computer stock.

A yKevin = 500x + 5000

yKeith = –125x2 + 2500x + 5000

B yKevin = 500x + 5000

yKeith = 125x2 + 2500x + 5000

C yKevin = 500x

yKeith = –125x2 + 2500x

D yKevin = 500x + 5000

yKeith = –5000x2 + 17500x + 155000

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56 Tommy was trying to solve the system of a linear equation and a quadratic equationshown below.

2x – y = –1

x2 – 5x – y = –7

Tommy had completed the following steps in his algebraic solution process but could

not factor the resulting quadratic, x2 – 7x – 1, to complete the problem.

Step I: y = 2x + 1

Step II: x2 – 5x – (2x + 1) = –7

Step III: x2 – 5x – 2x – 1 = –7

Step IV: x2 – 7x – 1 = –7

Determine the step in which Tommy made his mistake and justify your reasoning.

F Step I; Tommy should have transformed the first equation to y = 2x – 1.

G Step II; Tommy should have substituted the expression 2x + 1 in for eachvariable.

H Step III; Tommy should have distributed into the parentheses yielding the

equation x2 – 5x – 2x + 1 = –7.

J Step IV; Tommy must set the quadratic expression equal to 0 before factoring.

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