The graph of ax + by = c, dx + ey = f will be
I. parallel, if the system has no solution.
II. coincident, if the system has finite number of solutions.
III. intersecting. if the system has only one solution.
Which of the above statements are correct?

[?3 + x + ?3 - x] / [?3 + x - ?3 - x] = 2

Let ?3 + x = a and ?3 - x = b
Then, (a + b) / (a - b) = 2/1
? a + b = 2a - 2b
? a = 3b

? ?3 + x = 3?3 - x

On squaring bothv sides, we get
(?3 + x)² = (3?3 - x)²
? 3 + x = 9(3 - x)
? 3 + x = 27 - 9x
? 10x = 24
? x = 12/5

Given,
(x + y - 8)/2 = (x + 2y - 14)/3 = (3x + y - 12)/11
? (x + y - 8)/2 = (x + 2y - 14)/3
? 3x + 3y - 24 = 2x + 4y - 28
? 3x + 3y - 2x - 4y = -28 + 24
x - y = -4 ...(i)

Again, (x + 2y - 14)/3 = (3x + y - 12)/11
? 11x + 22y - 154 = 9x + 3y - 36
? 2x + 19y = 118 ..(ii)


On multiplying Eq. (i) by 2 and subtracting from Eq., we get
2x - 2y = -8
2x + 19y = 118
-------------------
-21y = -126
? y = 6

On putting the value of y in Eq. (i), we get
x - 6 = -4
? x = 2
? x = 2 and y = 6

Let the number of gold coins initially be x and the number of non-gold coins be y.
According to the question,
3x = y

When 10 more gold coins, total number of gold coins becomes x + 10 and the number of non-gold coins remain the same at y.
Now, we have 2(x + 10) = y

Solving these two equations, we get
x = 20 and y = 60.
Total number of coins in the collection at the end is equal to
x + 10 + y = 20 + 10 + 60 = 90.

Given equations 2x + 4y = 6 and 4x + 8y = 6

then,
a₁/a₂ = 2/4 = 1/2;
b₁/b₂ = 4/8 = 1/2;
c₁/c₂ = 6/6 = 1
? a₁/b₂ = b₁/b₂ ? c₁/c₂

So there is no solution for these equations.

Given, 6x - 10y = 10 ..........(i)
and x/(x + y) = 5/7
? 7x = 5x + 5y
? 2x - 5y = 0 ...(ii)

On multiplying Eq. (ii) by 2 and subtracting from Ed.(i), we get
6x - 10y = 10
4x - 10y = 0
---------------------
2x = 10
? x = 5
Putting the value of x in Eq. (i), we get
30 - 10y = 10
? 10y = 20
? y = 2
? (x - y) = 5 - 2 = 3

Let the cost of one chair be ? x
and cost of one table be ? y.
By given condition,
10x + 6y = 6200 ..(i)
and 3x + 2y = 1900
? 9x + 6y = 5700 ...(ii)

On subtracting Eq. (ii), we get
x = ? 500

From Eq (i),
5000 + 6y = 6200
? 6y = 1200
? y = ? 200

The cost of 4 chair and 5 tables
= 4x + 5y
= 2000 + 1000
= ? 3000

Given, 2^a + 3^b = 17
and 2^{a + 2} - 3^{b + 1} = 5
? 2² x 2² - 3^b x 3¹ = 5
? 4.2^a - 3.3^b = 5
Let 2^a = x amd 3^b = y
Then, x + y = 17 ...(i)
4x - 3y = 5 ...(ii)
On multiplying Eq. (i) by 3 and adding to Eq (ii), we get
3x + 3y = 51
4x - 3y = 5
------------------
7x = 56
? x = 8

On putting the value of x in Eq. (i), we get
8 + y = 17
? y = 9

Now, 2^a = x
? 2^a = 8 (2)³
? 3^b = y = 9
? 3^b = 3²
? b = 2
Hence, a = 3 and b = 2

(p/x) + (q/y) = m ..(i)
(q/x) + (p/y) = n ...(ii)

On multiply Eq (i) by q and Eq. (ii) by p and subtracting, we get
(pq/x) + (q²)/y = mq
(pq/x) + (p²)/y = np
-----------------------------------------
(q²/y) - (p²/y) = mq - np
? (q² - p²) = y(mq - np)
? y = (q² - p²)/mp - np = (p² - q²)/np - mq

Again, on multiplying Eq. (i) by p and Eq. (ii) by q and subtracting, we get
(p²/x) + (pq/y) = mp
(q²/x) + (pq/y) = nq
-----------------------------------------
(p²/x) - (q²/x) = mp - nq
? (p² - q²) = x (mp - nq)
? x = (p² - q²)/(mp - nq)
? x = (p² - q²)/(mp - nq)
and y = (p² - q²) / (np - mq)

Let the capital of one is x and that of another is y .

According to the question.
x + 100 = 2(y - 100)
x + 100 = 2y - 200
or x - 2y = -300 ...(i)

Again, according to the question
y + 10 = 6(x - 10)
? y + 10 = 6x - 10
? 6x - y = 70 ..(ii)

On multiplying Eq. (ii) by 2 and subtracting from Eq. (i) , we get
x - 2y = -300
12x - 2y = 140
----------------
-11x = -440
? x = 40

On putting the value of x in Eq. (i) , we get
40 - 2y = -300
? 2y = 340
? y = 170
So, their capital are ? 40 and ? 170.

Let the incomes of two persons be 8x and 5x and their expenditure be 2y and y , respectively.
? Saving = Income - Expenditure
? 1000 = 8x - 2y ...(i)
and 1000 = 5x - y ...(ii)


On multiplying Eq. (ii) by 2 and subtracting from Eq. . (i) , we get
8x - 2y = 1000
10x - 2y = 2000
----------------------
-2x = -1000
? x = 500

? Monthly incomes are
8x = ? 4000 and
5x = ? 2500

? Difference = ? 4000 - 2500 = ? 1500