 |
VOOZH | about |
|
VOOZH | about |
System of Equations is the combination of equations containing variables. For example, 2x + 3y = 5 and 3x + 2y = 10 are systems of equations. In this article, we will learn how to solve a System of Equations.
A system of equations is a collection of two or more equations involving the same set of variables. The objective is to find the values of these variables that satisfy all the equations simultaneously. Systems of equations are used in various fields, including mathematics, physics, engineering, economics, and more.
Example: A movie theater sells two types of tickets: adult tickets for $12 each and child tickets for $8 each. On a particular day, the theater sells a total of 150 tickets and earns $1,500 in revenue. How many adult tickets and child tickets were sold?
- Let x be the number of adult tickets sold.
- Let y be the number of child tickets sold.
Set up the system of equations:
The total number of tickets sold is 150: x + y = 150 . . . (i)
The total revenue is $1,500: 12x + 8y = 1500 . . . (ii)
Solve the system using the substitution method:
Solve Equation (i) for y:y = 150 − x
Substitute this expression for y in Equation (ii):12x + 8(150 − x) = 1500
Distribute and combine like terms:12x + 1200 − 8x = 1500
4x + 1200 = 1500
Solve for x:4x = 300
⇒ x = 75
Now, substitute x = 75 back into Equation (i) to find y:y = 150 − 75 = 75
Answer:
The theater sold 75 adult tickets and 75 child tickets.
Verification:
- Total tickets: 75 + 75 = 150 (Matches Equation (i))
- Total revenue: 12(75) + 8(75) = 900 + 600 = 1500 (Matches Equation (ii))
Some important concepts related to solving a system of equations are listed below:
Problem 1: A fruit seller sells apples and oranges. The price of 3 apples and 4 oranges is $22. The price of 2 apples and 3 oranges is $16.
Solution:
Let the price of one apple be x and the price of one orange be y.
We have the following system of equations:3x + 4y = 22 . . . (i)
2x + 3y = 16 . . . (ii)
Step 1: Multiply Equation (i) by 2 and Equation (ii) by 3 to eliminate x:
6x + 8y = 44 . . . (iii)
6x + 9y = 48 . . . (iv)
Step 2: Subtract Equation 3 from Equation 4:
(6x + 9y) − (6x + 8y) = 48 − 44 y = 4
Step 3: Substitute y = 4 into Equation (i):3x + 4(4) = 22
3x + 16 = 22
⇒ x = 6
⇒ x = 2
Answer: The price of one apple is $2, and the price of one orange is $4.
Problem2 :A father is 3 times as old as his son. In 10 years, he will be twice as old as his son.
Solution:
Let the father's current age be x and the son's current age be y.
We have the following system of equations:x = 3y . . . (i)
x + 10 = 2(y + 10) . . . (ii)
Step 1: Substitute Equation (i) into Equation (ii):
3y + 10 = 2(y + 10)3
3y + 10 = 2y + 20
Step 2: Solve for y:y = 10
Step 3: Substitute y = 10 into Equation (i):x = 3(10) = 30
Answer: The father is 30 years old, and the son is 10 years old.
Problem 3: Two trains are moving towards each other from two stations 300 miles apart. Train A moves at 60 mph and Train B at 40 mph.
Solution:
Let t be the time in hours when they meet.
We have the following equation: 60t + 40t = 300
Step 1: Combine like terms: 100t = 300
Step 2: Solve for t: t = 3 hours
Answer: The trains will meet in 3 hours.
Problem 4:A store sells two types of pens. One costs $2 each and the other $3 each. A customer buys a total of 15 pens for $37.
Solution:
Let the number of $2 pens be x and the number of $3 pens be y.
We have the following system of equations:
x + y = 15 . . . (i)
2x + 3y = 37 . . . (ii)
Step 1: Solve Equation (i) for y:y = 15 − x
Step 2: Substitute into Equation (ii):2x + 3(15 − x) = 37
2x + 45 - 3x = 37
Step 3: Solve for x: − x + 45 = 37
⇒ x = 8
Step 4: Find y:y = 15 − 8 = 7
Answer: The customer bought 8 $2 pens and 7 $3 pens.
Problem 5:A farmer has 100 feet of fencing and wants to enclose a rectangular area along the river, using the river as one side of the rectangle.
Solution:
Let the length of the rectangle be x and the width be y.
We have the following system of equations:2x + y = 100 . . . (i)
Step 1: Solve for y:y = 100 − 2xy = 100 - 2xy = 100 − 2x
Step 2: Express the area A as a function of x:A = x⋅y = x(100 − 2x) = 100x − 2x2
Step 3: To maximize A, find the vertex of the parabola:x = − b2/a = − 100/ − 4 = 25
Step 4: Find y:y = 100 − 2(25) = 50
Answer: The dimensions that maximize the area are 25 feet by 50 feet.
Problem 6:A coffee shop sells a blend of coffee made from two types of beans. Type A costs $5 per pound, and Type B costs $8 per pound. The shop sells a 10 - pound mixture for $65.
Solution:
Let the number of pounds of Type A be x and Type B be y.
We have the following system of equations:
x + y = 10 . . . (i)
5x + 8y = 65 . . . (ii)
Step 1: Solve Equation (i) for y:y = 10 − x
Step 2: Substitute into Equation (ii):5x + 8(10 − x) = 65
5x + 80 - 8x = 65
Step 3: Solve for x: − 3x + 80 = 65⇒ x = 5
Step 4: Find y:y = 10 − 5 = 5
Answer: The mixture contains 5 pounds of Type A and 5 pounds of Type B.
Problem 7:A person invests $10,000 in two accounts, one with a 5% interest rate and the other with a 7% interest rate. The total interest earned in a year is $640.
Solution:
Let the amount invested at 5% be x and the amount at 7% be y.
We have the following system of equations:
x + y = 10000 . . . (i)
0.05x + 0.07y = 640 . . . (ii)
Step 1: Solve Equation (i) for y:y = 10000 − xy = 10000 - xy = 10000 − x
Step 2: Substitute into Equation (ii):0.05x + 0.07(10000 − x) = 640
.05x + 700 − 0.07x = 640
Step 3: Solve for x: − 0.02x + 700 = 640⇒ x = 300
Step 4: Find y:y = 10000 − 3000 = 7000
Answer: $3000 was invested at 5%
Problem 1: A boat travels 30 miles upstream in 4 hours and returns downstream in 3 hours. What are the speed of the boat in still water and the speed of the current?
Problem 2: Two planes leave the same airport, one heading north at 450 mph and the other heading west at 600 mph. After 2 hours, how far apart are they?
Problem 3:A man buys 5 pencils and 3 pens for $2.50. The next day, he buys 2 pencils and 7 pens for $3.50. Find the cost of each pencil and pen.
Problem 4: Two cyclists start from the same point and travel in the same direction. One travels at 15 mph and the other at 12 mph. After how many hours will they be 9 miles apart?
Problem 5: A farmer has 50 chickens and cows. The total number of legs is 140. How many chickens and cows does he have?
Problem 6: A theater sells tickets for $10 and $15. If 300 tickets were sold and the total revenue was $3500, how many tickets of each type were sold?
Problem 7: A student has a collection of 100 coins consisting of nickels and dimes. The total value of the coins is $7.50. How many nickels and dimes does the student have?
Problem 8: A car rental company charges a flat fee of $50 and $0.20 per mile driven. If a customer pays $70, how many miles did they drive?
Problem 9: Two cars leave the same point, one traveling north at 60 mph and the other east at 80 mph. After 3 hours, how far apart are the cars?
Problem 10: A chemist needs to mix two solutions, one containing 10% acid and the other containing 20% acid, to make 100 liters of a solution containing 15% acid. How many liters of each solution should be used?
Answer 1: Speed of the boat in still water = 10 mph, Speed of the current = 2.5 mph
Answer 2: 1,170 miles apart
Answer 3: Cost of each pencil = $0.30, Cost of each pen = $0.40
Answer 4: 3 hours
Answer 5: 30 chickens and 20 cows
Answer 6: 200 tickets at $10, 100 tickets at $15
Answer 7: 50 nickels and 50 dimes
Answer 8: 100 miles
Answer 9: 300 miles apart
Answer 10: 50 liters of each solution
Read More,
