Samuel Dominic Chukwuemeka (SamDom For Peace)

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Chukwuemeka



Solved Examples - Translate 2 by 2 Linear System Word Problems

Something About Word Problems ☺

(1.) Word problems are written in standard British English.

(2.) Some word problems are very lengthy and some are unnecessary. Those ones are meant to discourage you from even trying.

(3.) Word problems in Mathematics demonstrate the real-world applications of mathematical concepts.

(4.) Embrace word problems. See it as writing from "English to Math".
Take time to:
(a.) Read to understand. Paraphrase and shorten long sentences as necessary.
(b.) Re-read and note/underline the vocabulary "math" terms written in English.
(c.) Translate/Write "important" sentences one at a time.
(d.) Review what you wrote to ensure correctness.
(e.) Solve the math, and check your solution in the word problem.
Does that solution makes sense?
If it does, you are probably correct.
If it does not, please re-do it.

For ACT Students
The ACT is a timed exam...$60$ minutes for $60$ questions
So, you should try to solve each question correctly and timely
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

Translate these word problems to Math.
Use appropriate variables.
Define your variables.
Do not solve unless you wish.

(1.) The sum of two numbers is $-4$.
The difference between the numbers is $28$.
Find the numbers.


Let the numbers be $x\: and\: y$

$ x + y = -4 ...eqn.(1) \\[3ex] x - y = 28 ...eqn.(2) $
(2.) In the City of Surprise, Arizona for a certain year; there were a total of 54 commercial and non-commercial orbital launches.
The number of non-commercial launches was two more than three times the number of commercial launches.
Determine the number of commercial and non-commercial orbital launches.


Let the number of commercial launches be $x$ and
the number of non-commercial launches be $y$

$ x + y = 54 ...eqn.(1) \\[3ex] y = 2 + 3x ...eqn.(2) $
(3.) The average apartment rent in the State of California is $1390 per month.
Jonah has an apartment in the City of San Francisco and another one in the City of Santa Monica.
The total monthly rent for the two apartments in $4802.
The rent in the City of San Francisco is $1116 less than the rent in the City of Santa Monica.
Calculate the rent for each apartment.


Let the rent in the City of Santa Monica be $x$
and the rent in the City of San Francisco = $y$

$ x + y = 4802 ...eqn.(1) \\[3ex] y = x - 1116 ...eqn.(2) $
(4.) The tens digit of a two-digit number is six less than seven times the unit digit.
The difference between the digits is six.
Find the numbers.


Let the two-digit number be $xy$
The tens digit = $x$
The unit digit = $y$

$ x = 7y - 6 ...eqn.(1) \\[3ex] x - y = 6 ...eqn.(2) $
(5.) Assume a plane is flying at a constant speed under unvarying wind conditions.
Traveling against a head wind, the plane takes 4 hours to travel 564 miles.
Traveling with a tail wind, the plane flies 549 miles in 3 hours.
Calculate the speed of the plane, and the speed of the wind.


This is too much grammar for some folks! Can we paraphrase?

$Speed = \dfrac{distance}{time}$

Speed of plane against the wind = $\dfrac{564}{4} = 141\: mph$

Speed of plane with the wind = $\dfrac{549}{3} = 183\: mph$

We can re-write that word problem as:
Flying against the wind, a plane flew $141\: mph$
Flying with the wind, a plane flew $183\: mph$
Calculate the speed of the plane, and the speed of the wind.

Let the speed of the plane = $x$
Let the speed of the wind = $y$

$ x - y = 141 ...eqn.(1) \\[3ex] x + y = 183 ...eqn.(2) $
(6.) Assume that Singapore and Paris are the two most expensive cities to live in 2014.
On the basis of the average cost per day for each city (which includes a hotel room, car rental, and three meals), 2 days in Singapore and 3 days in Paris cost $2772, while 4 days in Singapore and 2 days in Paris cost $3488.
What is the average cost per day in each city?


Let the:
average cost per day of living in Singapore be $x$
and the average cost per day of living in Paris be $y$

$ 2x + 3y = 2772 ...eqn.(1) \\[3ex] 4x + 2y = 3488 ...eqn.(2) $
(7.) Paul's Coffee charges $8 per pound for Kenyan French Roast coffee and $7 per pound for Sumatran coffee.
How much of each type should be used to make a 20 pound blend that sells for $7.65 per pound?


Two things are important here: Amount (pounds) and Cost (dollars)
Let the amount of Kenyan French Roast coffee be $x$
and the amount of Sumatran coffee be $y$
Amount
$x + y = 20 ...eqn.(1)$
Cost
Cost of Kenyan French Roast coffee @ $8 per pound for $x\: pounds = 8x$
Cost of Sumatran coffee @ $7 per pound for $y\: pounds = 7y$
Cost of the 20 pound blend @ $7.65 per pound for $20 * 7.65 = 153$
$8x + 7y = 153 ...eqn.(2)$

The two equations are:

$ x + y = 20 ...eqn.(1) \\[3ex] 8x + 7y = 153 ...eqn.(2) $
(8.) Judith invested $7000 between two accounts paying 7% and 9% annual interest, respectively.
If the total interest earned for the year was $590,
how much was invested at each rate?


Let the investment at 7% interest rate be at Bank A
Let that investment be $x$
Let the investment at 9% interest rate be at Bank B
Let that investment be $y$
Judith invested $7000 between two accounts $x + y = 7000 ...eqn.(1)$
Recall the formula for Simple Interest Simple Interest = Principal * Rate * Time
$I = P * r * t$
Bank A
$P = x$
$r = 7% = 0.07$
$t = 1$
$I = x * 0.07 * 1 = 0.07x$
Bank B
$P = y$
$r = 9% = 0.09$
$t = 1$
$I = y * 0.09 * 1 = 0.09y$
Total Interest from Bank A and Bank B = $0.07x + 0.09y$
The total interest earned for the year was $590
$0.07x + 0.09y = 590 ...eqn.(2)$

The two equations are:

$ x + y = 7000 ...eqn.(1) \\[3ex] 0.07x + 0.09y = 590 ...eqn.(2) $
(9.) An electronics store in the town of Experiment, Georgia took presale orders for a new smartphone and a new tablet.
There were 240 preorders for the smartphone and 170 preorders for the tablet.
The combined sale of the preorders is $252,250.
The price of a smartphone and a tablet together is $1,175
Calculate the cost of one smartphone and the cost of a tablet.


Let the cost of a smartphone be $x$
and the cost of a tablet be $y$
The price of a smartphone and a tablet together is $1,175
$x + y = 1175 ...eqn.(1)$
240 preorders of smartphones @ $x per smartphone = $240x$
170 preorders of tablets @ $y per tablet = $170y$
The combined sale of the preorders is $252,250.
$240x + 170y = 252250 ...eqn.(2)$

The two equations are:

$ x + y = 1175 ...eqn.(1) \\[3ex] 240x + 170y = 252250 ...eqn.(2) $
(10.) Onesimus was asked by a business firm to be a full partner or a sales manager.
As a full partner, he will receive an annual salary of $20,000 and 25% of the company's yearly profits.
As a sales manager, he will receive an annual salary of $30,000 and 20% of the company's yearly profits.
What must the yearly profit be for his total earnings to be the same, regardless of which option he chose?


Two factors are in question: Total earnings and Profits
Let the total earning be $x$
and the yearly profit be $y$
As a full partner: $x = 20000 + 0.25y ...eqn.(1)$
As a sales manager: $x = 30000 + 0.2y ...eqn.(2)$

The two equations are:

$ x = 20000 + 0.25y ...eqn.(1) \\[3ex] x = 30000 + 0.2y ...eqn.(2) $
(11.) A one-cup serving of spaghetti with meatballs contains 260 calories and 32 grams of carbohydrates.
A one-cup serving of chopped iceberg lettuce contains 5 calories and 1 gram of carbohydrates.
Determine how many servings of each would be required to obtain 320 calories and 40 grams of carbohydrates.


Two factors are important here: Calorie counts and Carbohydrates
Let the amount/servings of spaghetti meatballs = $x$
and the amount/servings of iceberg lettuce be $y$
Calories
Meatballs: x servings @ 260 calories / one-cup = $260x$
Lettuce: y servings @ 5 calories / one-cup = $5y$
From the question, 320 calories is needed
$260x + 5y = 320 ...eqn.(1)$
Carbohydrates
Meatballs: x servings @ 32 grams / one-cup = $32x$
Lettuce: y servings @ 1 gram / one-cup = $1y$
From the question, 40 grams of carbohydrates is needed
$32x + 1y = 40 ...eqn.(2)$

The two equations are:

$ 260x + 5y = 320 ...eqn.(1) \\[3ex] 32x + 1y = 40 ...eqn.(2) $
(12.) Felicity jogs and walks to campus each day.
She averages 3 km/hr walking and 6 km/hr jogging.
The distance from home to the campus is 5 km, and she makes the trip in 1 hr.
How far does she jog on each trip?


Let the time she walks = $x$
and the time she jogs = $y$
... she makes the trip in 1 hr.

$ x + y = 1 ...eqn.(1) \\[3ex] Speed = \dfrac{distance}{time} \\[5ex] $ So, $distance = speed * time$
Distance from Walking
$3km/hr * x = 3x$
Distance from Jogging
$6km/hr * y = 6y$
The distance from home to the campus is 5 km ...
$3x + 6y = 5 ...eqn.(2)$

The two equations are:

$ x + y = 1 ...eqn.(1) \\[3ex] 3x + 6y = 5 ...eqn.(2) $
(13.) Rita's Foods sells cashews for $6.00 per pound and peanuts for $2.50 per pound.
The manager decides to mix 10 pounds of peanuts with some cashews and sell the mixture for $4.00 per pound.
How many pounds of cashews should be mixed with peanuts so that the mixture will produce the same revenue as would selling the "nuts" separately?
Cashew is a nut.
Peanut is a nut.
... as would selling the "nuts" separately? means ... as would selling the cashew and peanut separately?


Two things are important here: Amount/Weight (pounds) and Revenue ($)
Let the amount of cashews be $x$
and the amount of the mixture be $y$
Amount
The manager decides to mix 10 pounds of peanuts with some cashews ...
$x + 10 = y ...eqn.(1)$
Revenue
Revenue = Amount * Cost
Revenue for cashews: $x * 6 = 6x$
Revenue for peanuts: $10 * 2.50 = 25$
Revenue for the mixture = $y * 4.00 = 4y$
... so that the mixture will produce the same revenue as would selling the peanuts separately?
$4y = 6x + 25 ...eqn.(2)$

The two equations are:

$ 10 + x = y ...eqn.(1) \\[3ex] 4y = 6x + 25 ...eqn.(2) $
(14.) Do you drink? Please do not get drunk! Timothy wants to obtain a 10 liters solution containing 40% alcohol.
He mixes a 30% alcohol solution with a 70% alcohol solution.
How many liters of each solution should he mix?


Here is my video on Mixture Problems
Two factors are here: Volume and Concentration.
Let the volume of the 30% alcohol solution be $x$
and the volume of the 70% alcohol solution be $y$
Volume
$x + y = 10 ...eqn.(1)$
Concentration
30% of $x$ + 70% of $y$ = 40% of the mixture

$ 0.3x + 0.7y = 0.4 * 10 \\[3ex] 0.3x + 0.7y = 4 ...eqn.(2) \\[3ex] $ The two equations are:

$ x + y = 10 ...eqn.(1) \\[3ex] 0.3x + 0.7y = 4 ...eqn.(2) $
(15.) Naomi, a recent retiree, wants $$6000$ per year in extra income.
She has $$70,000$ to invest.
She can invest in B-rated bonds that pays $15\%$ interest per year or in a certificate of deposit (CD) that pays $7\%$ per year.
How much money should she invest in each account if she wants to get exactly $$6000$ in interest per year?


Let the investment (principal) in B-rated bonds be $x$
Let the investment (principal) in CD be $y$
Naomi has a sum of $70000 for both investments.
Per year means $1$ year
$x + y = 70000 ...eqn.(1)$
Interest = Principal * Rate * Time
B-rated bonds
Interest, I = x * 15% * 1
$I = 0.15x$
Certificate of deposit (CD)
Interest, I = y * 7% * 1
$I = 0.07y$
She wants exactly $6000 in interest.
This means that the total interest (interest from both investments) should be $6000
$0.15x + 0.07y = 6000 ...eqn.(2)$

The two equations are:

$ x + y = 70000 ...eqn.(1)$ \\[3ex] 0.15x + 0.07y = 6000 ...eqn.(2) $
(16.) Mekus Farms has 300 acres of land allotted for cultivating rice and beans.
The cost of cultivating rice and beans (including seeds and labor) is $3 and $15 per acre, respectively.
Mekus has $1500 available for cultivating these crops.
He wishes to use all the allotted land and his entire budget for cultivating these two crops.
How many acres of each crop should he plant?


Two factors are important here: Land and Cost
Let the acre of rice = $x$
and the acre of beans = $y$
Mekus Farms has 300 acres of land ...
Land: $x + y = 300 ...eqn.(1)$
Cost for Rice
$x$ acres @ $3 per acre = $3x$
Cost for Beans
$y$ acres @ $15 per acre = $15y$
Mekus has $1500 available for cultivating ...
Cost: $3x + 15y = 1500 ...eqn.(2)$

The two equations are:

$ x + y = 300 ...eqn.(1) \\[3ex] 3x + 15y = 1500 ...eqn.(2) $
(17.) Dietician John recommends a daily intake of 70 mg of Vitamin C and 50 mg of Vitamin D.
The local pharmacy stocks two compounds that can be used.
One contains 40% Vitamin C and 50% of Vitamin D.
The other contains 40% of Vitamin C and 20% of Vitamin D
How many milligrams of each compound should be mixed to satisfy the dietician's recommendation?


Let the two compounds be Compound A and Compound B
Let the amount of Compound A be $x$.
and the amount of Compound B be $y$
Compound A
One contains 40% Vitamin C and 50% of Vitamin D.
Amount of Vitamin C in Compound A: 40% of $x = 0.4x$
Amount of Vitamin D in Compound A: 50% of $x = 0.5x$
Compound B
The other contains 40% of Vitamin C and 20% of Vitamin D.
Amount of Vitamin C in Compound B: 40% of $y = 0.4y$
Amount of Vitamin D in Compound B: 20% of $y = 0.2y$
Daily Intake of Vitamin C and Vitamin D (from both Compounds)
Dietician John recommends a daily intake of 70 mg of Vitamin C and 50 mg of Vitamin D.

The two equations are:

$ 0.4x + 0.4y = 70 ...eqn.(1) \\[3ex] 0.5x + 0.2y = 50 ...eqn.(2) $
(18.) Lebron Durant and Kevin James measured a basketball court.
They found that the width of the court was $34$ feet less than the length.
If the perimeter of the court was $350$ feet, calculate the length and width of the court.


Let the length of the court be $x$
and the width be $y$
The width of the court was 34 feet less than the length
$y = x - 34 ...eqn.(1)$
A basketball is rectangular in shape.
Perimeter of a rectangular = $2 * length + 2 * width$
$2x + 2y = 350 ...eqn.(2)$

The two equations are:

$ y = x - 34 ...eqn.(1) \\[3ex] 2x + 2y = 350 ...eqn.(2) $
(19.) Peter traveled upstream a distance of 56 miles at top speed to a fishing spot in 4 hours.
On his return, he noticed that his trip downstream, still at top speeed, takes 3.5 hours.
Calculate the rate of Peter's boat and the rate of the current.


$ Speed = \dfrac{distance}{time} \\[5ex] Upstream\: speed = \dfrac{56}{4} = 14 mph \\[5ex] Downstream\: speed = \dfrac{56}{3.5} = 16 mph \\[5ex] $ Ask students why the downstream speed is greater than the upstream speed
Explain the interdisciplinary concept in Physics.
Provide more examples of this concept/force of gravity
Let the rate of Peter's boat = $x$
and the rate of the current be $y$

$ \underline{Upstream}: x - y = 14 ...eqn.(1) \\[3ex] \underline{Downstream}: x + y = 16 ...eqn.(2) $
(20.) During a one-month promotional campaign, Peace Films gave either a free DVD rental or a 12-serving box of popcorn to new members.
It cost the firm $1 for each free rental and $2 for each box of popcorn.
In all, 51 new members were signed up and the firm's cost for the incentives was $93.
How many of each incentive were given away?


Let the number of free rentals be $x$
and the number of boxes of popcorn be $y$
Two variables are here: Number of Freebies and Cost
Number of Freebies
In all, 51 new members were signed up ...
$x + y = 51 ...eqn.(1)$
Cost
$1 for each free rental and $2 for each box of popcorn.
... and the firm's cost for the incentives was $93 $1x + 2y = 93 ...eqn.(2)$

The two equations are:

$ x + y = 51 ...eqn.(1) \\[3ex] x + 2y = 93 ...eqn.(2) $




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(21.) A certain grocery store in the town of Chicken, Alaska did not mark prices on its goods.
Cosmas bought four loaves of bread and five fish from the store at a cost of $17.61
Damian paid a sum of $19.11 for five loaves of bread and four fish at the same store.
They want to return four loaves of bread and four fish.
How much should be refunded to them?


Let the cost for a loaf of bread = $x$
and the cost for a fish be $y$
4 loaves of bread @ $x per loaf = $4x$
5 loaves of bread @ $x per loaf = $5x$
five fish @ $y per fish = $5y$
four fish @ $y per fish = $4y$
Spent by Cosmas
Cosmas bought four loaves of bread and five fish from the store at a cost of $17.61
$4x + 5y = 17.61 ...eqn.(1)$
Spent by Damian
Damian paid a sum of $19.11 for five loaves of bread and four fish at the same store.
$5x + 4y = 19.11 ...eqn.(2)$

The two equations are:

$ 4x + 5y = 17.61 ...eqn.(1) \\[3ex] 5x + 4y = 19.11 ...eqn.(2) $
(22.) Evelyn invested an amount of $2250 in two different banks.
She invested some amount in a bank that offers 6% interest rate, and the rest of the money in a bank that offers 8% interest rate.
The yearly income on the 8% investment was $6 more than twice the income from the 6% investment.
How much did she invest in each bank?


Let the banks be Bank A and Bank B
Let the investment (principal) in Bank A be $x$
and the investment (principal) in Bank B be $y$
Evelyn invested an amount of $2250 in two different banks.
Yearly means $1$ year
$x + y = 2250 ...eqn.(1)$
Interest = Principal * Rate * Time
Bank A
$ Interest,\:\: I = x * 6\% * 1 \\[3ex] I = 0.06x \\[3ex] $ Bank B
$ Interest,\:\: I = y * 8\% * 1 \\[3ex] I = 0.08y \\[3ex] $ The yearly income on the 8% investment was $6 more than twice the income from the 6% investment.

$ 0.08y = 6 + 2(0.06x) \\[3ex] 0.08y = 6 + 0.12x ...eqn.(2) \\[3ex] $ The two equations are:

$ x + y = 2250 ...eqn.(1) \\[3ex] 0.08y = 6 + 0.12x ...eqn.(2) $