Simultaneous Linear Equations — Competency Focused Questions

If 0.4x + 0.3y = 2.3 and 2.5x - 2y = -5, then the value of xy is :

1. 10

2. 12

3. 2.4

4. 1.2

Answer

Given,

0.4x + 0.3y = 2.3 and 2.5x - 2y = -5

Solving first equation,

⇒ 0.4x + 0.3y = 2.3

⇒ 10(0.4x + 0.3y = 2.3)     [Multiplying both sides by 10]

⇒ 4x + 3y = 23

⇒ 4x = 23 - 3y

⇒ x = $\dfrac{23 - 3y}{4}$     ....(1)

⇒ 2.5x - 2y = -5     ....(2)

Substituting value of x from equation (1) in 2.5x - 2y = -5, we get :

$\Rightarrow 2.5 \Big(\dfrac{23 - 3y}{4}\Big) - 2y = -5 \\[1em] \Rightarrow \dfrac{57.5 - 7.5y}{4} - 2y = -5 \\[1em] \Rightarrow \dfrac{57.5 - 7.5y - 8y}{4} = -5 \\[1em] \Rightarrow 57.5 - 15.5y = -5 \times 4 \\[1em] \Rightarrow 57.5 - 15.5y = -20 \\[1em] \Rightarrow -15.5y = -20 - 57.5 \\[1em] \Rightarrow -15.5y = -77.5 \\[1em] \Rightarrow y = \dfrac{-77.5}{-15.5} = 5.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{23 - 3y}{4} \\[1em] \Rightarrow x = \dfrac{23 - 3 \times 5}{4} \\[1em] \Rightarrow x = \dfrac{23 - 15}{4} \\[1em] \Rightarrow x = \dfrac{8}{4} = 2.$

x = 2 and y = 5.

xy = 10.

Hence, option 1 is the correct option.

If ax - by = a² + b² and $\dfrac{x + y}{2} = a$, then the value of x - y is :

1. a

2. 2a

3. b

4. 2b

Answer

Given,

Equations: ax - by = a² + b² and $\dfrac{x + y}{2} = a$

Solving equation $\dfrac{x + y}{2} = a$,

$\Rightarrow \dfrac{x + y}{2} = a \\[1em] \Rightarrow x + y = a \times 2 \\[1em] \Rightarrow x + y = 2a \\[1em] \Rightarrow y = 2a - x \text{ ......(1) }$

⇒ ax - by = a² + b²     .......(2)

Substituting value of y from equation (1) in (2) we get,

⇒ ax - by = a² + b²

⇒ ax - b(2a - x) = a² + b²

⇒ ax - 2ab + bx = a² + b²

⇒ ax + bx = a² + b² + 2ab

⇒ x(a + b) = a² + b² + 2ab

⇒ x(a + b) = (a + b)²

⇒ x = $\dfrac{(a + b)^2}{a + b}$

⇒ x = (a + b).

Substituting value of x in equation (1),

⇒ y = 2a - x

⇒ y = 2a - (a + b)

⇒ y = 2a - a - b

⇒ y = a - b

Now,

⇒ x - y = (a + b) - (a - b)

⇒ x - y = (a + b - a + b)

⇒ x - y = 2b.

Hence, option 4 is the correct option.

If 8x + 9y = 42xy and 2x + 3y = 12xy, then the value of $\dfrac{1}{xy}$ is :

1. 1

2. 4

3. 6

4. $\dfrac{1}{6}$

Answer

Given,

Equations:

⇒ 8x + 9y = 42xy

⇒ 2x + 3y = 12xy

Dividing both the sides of first equation by xy, we get :

$\Rightarrow \dfrac{8x + 9y}{xy} = \dfrac{42xy}{xy} \\[1em] \Rightarrow \dfrac{8x}{xy} + \dfrac{9y}{xy} = 42 \\[1em] \Rightarrow \dfrac{8}{y} + \dfrac{9}{x} = 42 \text{ .......(1)}$

Dividing both the sides of second equation by xy, we get :

$\Rightarrow \dfrac{2x + 3y}{xy} = \dfrac{12xy}{xy} \\[1em] \Rightarrow \dfrac{2x}{xy} + \dfrac{3y}{xy} = 12 \\[1em] \Rightarrow \dfrac{2}{y} + \dfrac{3}{x} = 12.$

Multiplying both sides of the above equation by 4, we get :

$\Rightarrow 4\Big(\dfrac{2}{y} + \dfrac{3}{x}\Big) = 12 \times 4 \\[1em] \Rightarrow \dfrac{8}{y} + \dfrac{12}{x} = 48 \text{ .......(2)}$

Subtracting equation (1) from (2), we get:

$\Rightarrow \dfrac{8}{y} + \dfrac{12}{x} - \Big(\dfrac{8}{y} + \dfrac{9}{x}\Big) = 48 - 42 \\[1em] \Rightarrow \dfrac{8}{y} + \dfrac{12}{x} - \dfrac{8}{y} - \dfrac{9}{x} = 6 \\[1em] \Rightarrow \dfrac{12}{x} - \dfrac{9}{x} = 6 \\[1em] \Rightarrow \dfrac{12 - 9}{x} = 6 \\[1em] \Rightarrow \dfrac{3}{x} = 6 \\[1em] \Rightarrow x = \dfrac{3}{6} = \dfrac{1}{2}.$

Substituting value of x in equation (1), we get:

$\Rightarrow \dfrac{8}{y} + \dfrac{9}{x} = 42 \\[1em] \Rightarrow \dfrac{8}{y} + \dfrac{9}{\dfrac{1}{2}} = 42 \\[1em] \Rightarrow \dfrac{8}{y} + 18 = 42 \\[1em] \Rightarrow \dfrac{8}{y} = 42 - 18 \\[1em] \Rightarrow \dfrac{8}{y} = 24 \\[1em] \Rightarrow y = \dfrac{8}{24} = \dfrac{1}{3}.$

Calculating xy,

xy = $\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}$.

$\therefore \dfrac{1}{xy} = 6$

Hence, option 3 is the correct option.

A shopkeeper sold a table and a chair for ₹ 1,050, thereby making a profit of 10% on the table and 25% on the chair. If he had taken a profit of 25% on the table and 10% on the chair, then he would have got ₹ 1,065. What is the cost price of 1 table and 1 chair.

Answer

Let cost price of the table be ₹ x and cost price of the chair be ₹ y.

According to case 1 :

⇒ Profit on table = 10%

Selling Price of table = Cost price (1 + Profit%) = $x \Big(1 + \dfrac{10}{100}\Big)$ = ₹ x × 1.10

⇒ Profit on chair = 25%

Selling Price of chair = Cost price (1 + Profit%) = $y \Big(1 + \dfrac{25}{100}\Big)$ = ₹ y × 1.25

⇒ 1.10x + 1.25y = 1050

Multiply the equation by 100,

⇒ 100(1.10x + 1.25y) = 100 × 1050

⇒ 110x + 125y = 105000

⇒ 5(22x + 25y) = 5 × 21000

⇒ 22x + 25y = 21000

⇒ 22x = 21000 - 25y

⇒ x = $\dfrac{21000 - 25y}{22}$     ......(1)

According to case 2 :

⇒ Profit on table = 25%

Selling Price of table = Cost price (1 + Profit%) = $x \Big(1 + \dfrac{25}{100}\Big)$ = ₹ x × 1.25

⇒ Profit on chair = 10%

Selling Price of chair = Cost price (1 + Profit%) = $y \Big(1 + \dfrac{10}{100}\Big)$ = ₹ y × 1.10

⇒ 1.25x + 1.10y = 1065

Multiply the equation by 100,

⇒ 100(1.25x + 1.10y) = 1065 × 100

⇒ 125x + 110y = 106500

⇒ 5(25x + 22y) = 5 × 21300

⇒ 25x + 22y = 21300     .......(2)

Substituting value of x from equation 1 in (2), we get :

$\Rightarrow 25 \Big(\dfrac{21000 - 25y}{22}\Big) + 22y = 21300 \\[1em] \Rightarrow \dfrac{25(21000 - 25y) + 484y}{22} = 21300 \\[1em] \Rightarrow 525000 - 625y + 484y = 468600 \\[1em] \Rightarrow -141y = 468600 - 525000 \\[1em] \Rightarrow -141y = -56400 \\[1em] \Rightarrow y = \dfrac{-56400}{-141} = 400.$

Substituting value of y in equation 1, we get :

$\Rightarrow x = \dfrac{21000 - 25y}{22} \\[1em] \Rightarrow x = \dfrac{21000 - 25 \times 400}{22} \\[1em] \Rightarrow x = \dfrac{21000 - 10000}{22} \\[1em] \Rightarrow x = \dfrac{11000}{22} = 500.$

Hence, cost Price of Table = ₹ 500 and cost Price of Chair = ₹ 400.

A boatman rowing at the rate of 5 km/hr in still water takes thrice as much time in going 40 km upstream as in going 40 km downstream. What is the speed of the stream?

Answer

Let x be speed of the stream.

Given,

Speed of boat in still water = 5 km/hr.

Speed of boat in upstream = (5 - x) km/hr

Speed of boat in downstream = (5 + x) km/hr

By formula,

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

Given,

The boatman takes thrice as much time in going 40 km upstream as in going 40 km downstream.

$\therefore \dfrac{40}{5 - x} = 3 \times \dfrac{40}{5 + x} \\[1em] \Rightarrow \dfrac{1}{5 - x} = \dfrac{1}{5 + x} \\[1em] \Rightarrow 5 + x = 3(5 - x) \\[1em] \Rightarrow 5 + x = 15 - 3x \\[1em] \Rightarrow x + 3x = 15 - 3 \\[1em] \Rightarrow 4x = 10 \\[1em] \Rightarrow x = \dfrac{10}{4} \\[1em] \Rightarrow x = 2.5 \text{ km/hr }$

Hence, the speed of stream = 2.5 km/hr.

A shopkeeper buys pens and pencils at ₹ 5 and ₹ 1 per price respectively. For every two pens, he buys three pencils. He sold pens and pencils at 12% and 10% profit respectively. If his total sale is ₹ 725, then find the number of pens and pencils sold by him.

Answer

Let x be the number of pens sold and y be the number of pencils sold.

Given,

Cost Price (CP) of 1 pen = ₹ 5

Profit on pens = 12%

⇒ SP of 1 pen = CP of pen + (Profit % of CP)

⇒ SP of 1 pen = 5 + $\dfrac{12}{100} \times 5$ = 5 + 0.12 × 5 = 5 + 0.60 = ₹ 5.60

Given,

Cost Price (CP) of 1 pencil = ₹ 1

Profit on pencils = 10%

SP of 1 pencil = CP of pencil + (Profit % of CP)

SP of 1 pencil = 1 + $\dfrac{10}{100}$ × 1 = 1 + 0.10 = ₹ 1.10

For every two pens, he buys three pencils,

This means the ratio of pens to pencils is x : y = 2 : 3.

$\dfrac{x}{y} = \dfrac{2}{3}$

⇒ y = $\dfrac{3}{2}x$     .......(1)

Given,

Total sale is ₹ 725,

⇒ x × 5.60 + y × 1.10 = 725

⇒ 5.60x + 1.10y = 725     ......(2)

Substitute the expression for y from equation (1) in (2), we get :

⇒ 5.60x + 1.10 $\times \dfrac{3}{2}x$ = 725

⇒ 5.60x + 1.10 × 1.5 = 725

⇒ 5.60x + 1.65x = 725

⇒ 7.25x = 725

⇒ y = $\dfrac{725}{7.25}$ = 100.

Substituting value of y in equation 1 we get,

⇒ y = $\dfrac{3}{2} \times 100$

⇒ y = 3 × 50

⇒ y = 150.

Hence, the number of pens sold = 100 and number of pencils sold = 150.

The angles of a triangle in ascending order are x, y and z. If y - x = z - y = 10°, then find the angles of the triangle.

Answer

Given,

The three angles of the triangle in ascending order are x, y, and z.

Given,

⇒ y − x = 10°

⇒ y = 10° + x     ....(1)

Given,

⇒ z − y = 10°     ....(2)

Substitute value of y from equation (1) in (2), we get :

⇒ z − y = 10°

⇒ z - (10° + x) = 10°

⇒ z = 10° + 10° + x

⇒ z = 20° + x.

So, the three angles of the triangle in terms of x are:

First angle: x

Second angle: x + 10°

Third angle: x + 20°

We know that,

The sum of the angles in any triangle is always 180°.

⇒ x + y + z = 180°

⇒ x + (x + 10°) + (x + 20°) = 180°

⇒ 3x + 30° = 180°

⇒ 3x = 180° − 30°

⇒ 3x = 150°

⇒ x = $\dfrac{150°}{3}$

⇒ x = 50°.

Substituting the value of x,

⇒ y = x + 10° = 50° + 10° = 60°

⇒ z = x + 20° = 50° + 20° = 70°.

Hence, x = 50°, y = 60°, z = 70°.