Question: User: UNIT – I
Theory of Equations
Simultaneous Equations – Solve Using Elimination Method.
1. X+2Y =7; 2X- Y =4
2. 3X+7Y = 13; 5X-2Y=8
3. 2X-3Y=19; 3X+2Y=9
4. 3X+5Y+7= 0; 4X+7Y+3 =0
5. 5X-4Y =0; 9X-9Y =-9
Solve using Substitution Method
6. X+2Y=4; 3X+Y=7
7.3X+4Y=2; 5X+7Y=0
8. 8X-3Y =3; 6X-6Y= 1
9. 4(1-P) = 7Q+8P; 6P+Q+8 =0
10. X=5y-3; 3X-8Y=12
11. At a theatre, 5 adult tickets and 3 child tickets cost ₹390, while 3 adult tickets and 4 child tickets cost ₹300. Find the price of an adult ticket and a child ticket.
12. 30Kgs of Commodity A and 25 Kgs of Commodity B together cost Rs.7100 and 25Kgs of Commodity A and 13 Kgs of Commodity B together cost Rs.5050. Find the price of Commodity A and B.
13. There are two numbers such that 3 times the first, twice the other is added the sum is 72. Also, if from 5 times the first number, 3times the other is subtracted, the result is 44. What are the numbers?
14. One number is greater by 6 than twice another number, but 3 times the smaller number exceeds the greater by one. Find the numbers.
15. The mobile phone charges per month of a cell phone operator can be expressed in the form c= a+nb where 9c is the bill amount, 9a is the fixed charges, 9b is variable charge per call and 9n is the number of calls made. When a user made 104 calls he received a bill for Rs.583 and when the number of calls was 67 the bill was Rs.509. What is the fixed charge and cost of each call?
1. **Problem:** Solve the simultaneous equations using the elimination method:
$$X + 2Y = 7$$
$$2X - Y = 4$$
2. **Formula and Rules:**
To solve by elimination, multiply equations to align coefficients of one variable, then add or subtract to eliminate that variable.
3. **Step 1:** Multiply the first equation by 1 and the second by 2 to align $Y$ coefficients:
$$X + 2Y = 7$$
$$4X - 2Y = 8$$
4. **Step 2:** Add the two equations to eliminate $Y$:
$$X + 2Y + 4X - 2Y = 7 + 8$$
$$5X = 15$$
5. **Step 3:** Solve for $X$:
$$X = \frac{15}{5} = 3$$
6. **Step 4:** Substitute $X=3$ into the first equation:
$$3 + 2Y = 7$$
$$2Y = 7 - 3 = 4$$
7. **Step 5:** Solve for $Y$:
$$Y = \frac{4}{2} = 2$$
**Final answer:**
$$X = 3, \quad Y = 2$$