Lesson 5 – Applications of Linear Equations: Part 3
Sa nakaraang aralin ay nai-apply natin ang ating natutunang konsepto at pamamaraan sa paglutas ng mga word problems na kinasasangkutan ng number, digit, uniform motion, money, age, investment, at mixture problems na may isang variable o unknown.
Sa bahaging ito ay pagtutunan naman natin ang pagsagot ng mga word problems gamit ang dalawang variables at/o dalawang equations.
Number Problems
May mga word problems na nalulutas sa pamamagitan lamang ng isang variable o unknown at isang equation. Gayunpaman, may mga pagkakataon na mas madali ang pagsagot ng mga ito kung gagamit ng dalawang variables at dalawang sistema ng equations.
Tunghayan natin sa mga sumusunod na halimbawa:
HALIMBAWA 1
The sum of the two numbers is 17 and their difference is 7. Find the two numbers.
Let x = the smaller number
y = the larger number
Given: x + y = 17 (Equation 1)
y – x = 7 (Equation 2)
Add the two equations.
x + y = 17
+ y – x = 7
2y = 24
y = 12
Substitute the value of y into either Equation 1 or 2 to find x.
y - x = 7 (Equation 2)
12 – x = 7
12 – 12 – x = 7 – 12
-x = -5
-1( -x = -5)
x = 5
Check our answer.
For Equation 1:
x + y = 17
5 + 12 ≟ 17
17 = 17
For Equation 2:
y - x = 7
12 - 5 ≟ 7
7 = 7
Thus, our smaller number is 5 and the larger number is 7.
REMEMBER
We can solve many problems by translating them into systems of equations and using the following problem-solving guidelines:
1. Understand the problem. Read it carefully and decide which quantities are unknown. (Unawain ang problema. Basahin itong mabuti at magpasya kung aling mga quantities ang hindi alam o unknown.)
2. Develop a plan. Represent one of the unknown values by one variable and the second unknown by another variable. (Bumuo ng isang plano. Katawanin ang isa sa mga unknown ng isang variable at ang pangalawa ng isa pang variable.)
3. Carry out your plan. Study the stated facts until you understand their meanings. Then translate the related facts into equations in two variables. Solve the system of equations. (Isagawa ang iyong plano. Pag-aralan ang mga nakasaad na katotohanan hanggang sa maunawaan mo ang mga kahulugan nito. Pagkatapos ay isalin ang mga kaugnay na katotohanan sa mga equation sa dalawang variable. Lutasin ang sistema ng mga equation.)
4. If available, check your answers using the derivations you made and not the given equation itself. Write a statement to answer the question being asked in the problem. (Kung mayroon, suriin ang iyong mga sagot gamit ang mga derivasyon na iyong ginawa at hindi ang ibinigay na equation mismo. Sumulat ng isang pahayag upang masagot ang tanong sa problema.)
Age Problems
HALIMBAWA 2
Olive is 4 years younger than Popeye. Twenty years ago, Popeye’s age was 13 years more than half the age of Olive. How old are they now?
Let x = Olive's age now
y = Popeye's age now
Labis na makatutulong kung ilalagay natin sa isang table ang mga ibinigay na impormasyon.
From the table, we can get a system of two equations in x and y.
x = y – 4 (Equation 1)
y – 20 = ½(x – 20) + 13 (Equation 2)
Simplifying Equation 2, we will have:
y – 20 = ½(x – 20) + 13 (Equation 2)
y – 20 = 1/2x – 10 + 13
y – 20 = 1/2x + 3
y = 1/2x + 23 (Equation 3
Substitute the value of x in Equation 1 into Equation 3.
y = ½(y – 4) + 23 (Equation 3)
y = 1/2y – 2 + 23
1/2y = 21
2(1/2y = 21)
y = 42
Substitute 42 as the value of y in Equation 1 to find the value of x.
x = y – 4 (Equation 1)
x = 42 – 4
x = 38
Thus, Olive is 38 years old and Popeye is 42 years old now.
Check our answer.
For Equation 3:
y = ½(y – 4) + 23 (Equation 3)
42 ≟ ½(42 – 4) + 23
42 ≟ ½(38) + 23
42 ≟ 19 + 23
42 = 42
For Equation 2:
y – 20 = ½(x – 20) + 13
42 – 20 ≟ ½(38 – 20) + 13
22 ≟ 19 – 10 + 13
22 = 22
Since we got equality, our answer is correct.
Digit Problems
HALIMBAWA 3
In a three-digit number, the hundreds digit is twice the units digit. If 396 be subtracted from the number, the order of the digits will be reversed. Find the number if the sum of the digits is 17.
Let h = the hundreds digit
t = the tens digit
u = the units digit
100h + 10t + u = the number
h = 2u (Equation 1)
h + t + u = 17 (Equation 2)
(100h + 10t + u) – 396 = 100u + 10t + h
100h + 10t + u – 396 -100u – 10t – h = 396
99h – 99u = 396
99h/99 - 99u/99 = 396/99
h – u = 4 (Equation 3)
Substitute h = 2u into Equation 3.
h – u = 4
2u – u = 4
u = 4
Substitute 4 as u into Equation 1.
h = 2u
h = 2(4)
h = 8
Substitute u = 8 and h = 4 into Equation 2 to find the value of t.
h + t + u = 17
4 + t + 8 = 17
4 – 4 + t 8 – 8 = 17 – 4 – 8
t = 5
Substitute these 3 values into our number.
100h + 10t + u =
100(8) + 10(5) + 4 =
800 + 50 + 4 = 854
Thus, our number is 854.
Let us check our answer by substituting our values into our three equations.
For Equation 1:
h = 2u
8 ≟2 (4)
8 = 8
For Equation 2:
h + t + u = 17
8 + 5 + 4 ≟ 17
17 = 17
For Equation 3:
h – u = 4
8 – 4 ≟ 4
4 = 4
Since our left and right terms for our three equations are balanced, our answer is correct.
Mixture Problems
HALIMBAWA 4
Tikyo wants to make a 1000 ml of 50% alcohol solution mixing a quantity of a 20% alcohol solution with a 70% alcohol solution. What are the quantities of each of the two solutions he has to use?
Let x = amount of 20% alcohol
y = amount of 70% alcohol
Analyzing the problem, we find that two conditions must be met.
Condition 1
liters of 20% solution + liters of 70% solution = liters of 50% solution
x + y = 1000
Condition 2
pure alcohol in 20% solution + pure alcohol in 70% solution = pure alcohol in 50% solution
0.20x + 0.70y = 0.50(1000)
The system we must solve is:
x + y = 1000 (Equation 1)
0.20x + 0.70y = 0.50(1000) (Equation 2)
Simplify Equation 2.
0.20x + 0.70y = 0.50(1000)
0.20x + 0.70y = 500
10(0.20x + 0.70y = 500)
2x + 7y = 5000 (Equation 3)
Multiply Equation 1 by 2.
x + y = 1000
2( x + y = 1000)
2x + 2y = 2000 (Equation 4)
Subtract Equation 4 from Equation 3.
2x + 7y = 5000 (Equation 3)
- (2x + 2y = 2000) (Equation 4)
5y = 3000
5y/5 = 3000/5
y = 600
Substitute 600 as the value of y into Equation 1.
x + y = 1000
x + 600 = 1000
x + 600 – 600 = 1000 – 600
x = 400
Thus, Tikyo has to use 400 liters of 20% alcohol and 600 liters of 70% alcohol to obtain 1000 liters of 50% alcohol.
Let us check to confirm our answer.
x + y = 1000 (Equation 1)
400 + 600 ≟ 1000
1000 = 1000
0.20x + 0.70y = 0.50(1000) (Equation 2)
10(0.20x + 0.70y ≟ 500)
2x + 7y ≟ 5000
2(400)x + 7(600) ≟ 5000
800 + 4200 ≟ 5000
5000 = 5000
Since we got equality in our original equations, our answer is correct.
Money Problems
HALIMBAWA 5
The price of 3 chairs and 2 tables is ₱4500 and the price of 5 chairs and 3 tables is ₱7000. Find the price of 2 chairs and 2 tables.
Let c = the price of a chair
t = the price of a table
Given:
3c + 2t = 4500 (Equation 1)
5c + 3t = 7000 (Equation 2)
Subtract Equation 1 from Equation 2.
5c + 3t = 7000 (Equation 2)
- (3c + 2t = 4500) (Equation 1)
2c + t = 2500
2c – 2c + t = 2500 – 2c
t = 2500 – 2c (Equation 3)
Substitute Equation 3 into either Eq’n 1 or 2.
If Equation 2:
5c + 3t = 7000
5c + 3(2500 – 2c) = 7000
5c + 7500 – 6c = 7000
-c + 7500 – 7500 = 7000 – 7500
-c = -500
-1(-c = -500)
c = 500
Substitute 500 for c into Equation 3 to find t.
t = 2500 – 2c (Equation 3)
t = 2500 – 2(500)
t = 2500 – 1000
t = 1500
Thus, the price of two chairs (2 x 500 = 1000) and two tables (2 x 1500 = 3000) is ₱ 4000 ( 1000 + 3000).
Let us check to confirm our answer.
3c + 2t = 4500 (Equation 1)
3(500) + 2(1500) ≟ 4500
1500 + 3000 ≟ 4500
4500 = 4500
5c + 3t = 7000 (Equation 2)
5(500) + 3(1500) ≟ 7000
2500 + 4500 ≟ 7000
7000 = 7000
t = 2500 – 2c (Equation 3)
1500 ≟ 2500 – 2(500)
1500 ≟ 2500 – 1000
1500 = 1500
Since we got equality in our original equations, our answer is correct.
Investment Problems
HALIMBAWA 6
Bongbong invested ₱11,000. Part of his money is invested in bonds which yield 8% and the remainder is invested in the money market which yields 10%. His total annual income from these investments is ₱1,020. Find the amount he has invested in each kind of investment.
Let b = amount he invested in bonds at 8%
m = amount he invested in money market at 10%
0.08b = annual income from 8% bonds investment
0.10m = annual income from 10% money market investment
b + m = 11000 (Equation 1)
0.08b + 0.10m = 1020 (Equation 2)
Multiply Equation 2 by 100.
0.08b + 0.10m = 1020 (Equation 2)
100(0.08b + 0.10m = 1020)
8b + 10m = 102000 (Equation 3)
Multiply Equation 1 by 8.
b + m = 11000 (Equation 1)
8(b + m = 11000)
8b + 8m = 88000 (Equation 4)
Subtract Equation 4 from Equation 3.
8b + 10m = 102000 (Equation 3)
- (8b + 8m = 88000) (Equation 4)
2m = 14000
2m/2 = 14000/2
m = 7000
Substitute 7000 as the value of m into Equation 1 to find b.
b + m = 11000 (Equation 1)
b + 7000 = 11000
b + 7000 – 7000 = 11000 – 7000
b = 4000
Thus, Bongbong invested ₱4000 in bonds at 8% and ₱7000 in money market at 10%.
Let us check to confirm our answer.
b + m = 11000 (Equation 1)
4000 + 7000 ≟ 11000
11000 = 11000
0.08b + 0.10m = 1020 (Equation 2)
100(0.08b + 0.10m ≟ 1020)
8b + 10m ≟ 102000
8(4000) + 10(7000) ≟ 102000
32000 + 70000 ≟ 102000
102000 = 102000
Since we got equality in our original equations, our answer is correct.
Motion Problems
HALIMBAWA 7
A boat can travel 16 miles up a river in 2 hours. The same boat can travel 36 miles downstream in 3 hours. What is the speed of the boat in still water? What is the speed of the current?
Let s = speed of the boat in still water
v = speed of the current
Kung hindi pa bihasa sa ganitong klaseng problema, makatutulong nang malaki ang paggawa ng table upang ilagay ang ating mga datos.
Mapapansin na ang rate (s + v) ng boat downstream ay ang speed nito sa kalmang tubig (still water) PLUS ang speed ng current (o daloy ng tubig) dahil mas mabilis ang takbo ng bangka kung paibaba o paayon sa daloy ng tubig. Kung salungat naman sa daloy ng tubig o upstream, magiging mabagal ang takbo ng bangka kaya ibabawas natin ang speed ng current sa kanyang speed sa still water upang makuha ang rate (s - v) nito.
Mula sa talahanayan o table, narito ang ating nakuhang equation:
16 = 2(s - v) (Equation 1)
36 = 3(s + v) (Equation 2)
Divide Equation 1 by 2.
16 = 2(s - v) (Equation 1)
16/2 = 2(s - v)/2
8 = s - v (Equation 3)
Divide Equation 2 by 3.
36 = 3(s + v) (Equation 2)
36/3 = 3(s + v)/3
12 = s + v (Equation 4)
Add Equation 3 and Equation 4.
8 = s - v (Equation 3)
+ 12 = s + v (Equation 4)
20 = 2s
20/2 = 2s/2
10 = s
Substitute 10 for s in any of the equations to find v. Let’s take Equation 4.
12 = s + v (Equation 4)
12 = 10 + v
12 – 10 = 10 – 10 + v
2 = v
Thus, the speed of the boat in still water is 10 miles per hour while the speed of the current is 2 miles per hour.
Let us check to confirm if our answer is correct.
16 = 2(s - v) (Equation 1)
16 ≟ 2(10 – 2)
16 ≟ 2(8)
16 = 16
36 = 3(s + v) (Equation 2)
36 ≟ 3(10 + 2)
36 ≟ 3(12)
36 = 36
Since we got equality in our original equations, our answer is correct.
Pagsasanay
Solve the following word problems and check your answers:
1. If one number is three times as large as another number and the smaller number is increased by 19, the result is 6 less than twice the larger number. What is the larger number?
2. A silver coin is 28 years older than a bronze coin. In 6 years, the silver coin will be twice as old as the bronze coin. Find the present age of each coin.
3. The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.
4. You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?
5. The cost of admission to a popular music concert was ₱162 for 12 children and 3 adults. The admission was ₱122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?
6. Isko has invested in two savings accounts. One earns 10% and the other earns 15%. He invests ₱200 more in the account that earns 15%. The total interest earned for one year is ₱230. How much is invested in each account?
7. A steamer goes downstream and covers the distance between two ports in 4 hrs., while it covers the same distance upstream in 5 hrs. If the speed of the stream is 2km/h, find the speed of the steamer in still water.
8. The present ages of Bongbong and Sara are in the ratio 3:4. Five years from now, the ratio of their ages will be 4:5. Find their present ages.
Note: The above sample word problems were taken from different internet sites.
ANSWERS:
