LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

3.1 GRAPHICAL SOLUTION OF LINEAR EQUATIONS IN TWO VARIABLES :

Graphs of the type (i) $a x=b$

Ex. 1 Draw the graph of following :

(i) $x=2$, (ii) $2 x=1$ (iii) $x+4=0$ (iv) $x=0$

(i) $x=2$

LINEAR EQUATIONS IN TWO VARIABLES II

(ii) $2 x=1 \Rightarrow x=\frac{1}{2}$

LINEAR EQUATIONS IN TWO VARIABLES II

(iii) $x+4=0 \Rightarrow x=-4$

LINEAR EQUATIONS IN TWO VARIABLES II

(iv) $x=0$

Graphs of the type (ii) ay $=b$.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 2 Draw the graph of following : (i) $y=0$,$\quad$ (ii) $y-2=0$, $\quad$ (iii) $2 y+4=0$

(i) $y=0$

LINEAR EQUATIONS IN TWO VARIABLES II

(ii) $y-2=0$

LINEAR EQUATIONS IN TWO VARIABLES II

(iii) $2 y+4=0 \Rightarrow y=-2$

Graphs of the type (iii) ax + by $=0$ (Passing through origin)

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 3 Draw the graph of following : (i) $x=y$ $\quad$ (ii) $x=-y$

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. (i) $x-y$

(ii) $x=-y$

LINEAR EQUATIONS IN TWO VARIABLES II

Graph of the Type (iv) ax+by+c=0.(Making interception x-axis , y-axis)

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 4 Solve the following system of linear equations graphically : $x-y=1,2 x+y=8$. Shade the area bounded by these two lines and $y$-axis. Also, determine this area.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol.

(i) $ x-y=1$

$x-y+1$

LINEAR EQUATIONS IN TWO VARIABLES II

(ii) $2 x+y=8$

$y=8-2 x$

Solution is $x=3$ and $y=2$

Area of is $x=3$ and $y=2$

Area of $\triangle ABC=\frac{1}{2} \times BC \times AD$

$=\frac{1}{2} \times 9 \times 3=13.5$ Sq. unit.

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

3.2 NATURE OF GRAPHICAL SOLUTION :

Let equations of two lines are $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$.

(i) Lines are consistent (unique solution) i.e. they meet at one point condition is $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

LINEAR EQUATIONS IN TWO VARIABLES II

(ii) Lines are inconsistent (no solution) i.e. they do not meet at one point condition is $\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

LINEAR EQUATIONS IN TWO VARIABLES II

(iii) Lines are coincident (infinite solution) i.e. overlapping lines (or they are on one another) condition is

$ \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} $

LINEAR EQUATIONS IN TWO VARIABLES II

3.3 WORD PROBLEMS:

For solving daily - life problems with the help of simultaneous linear equation in two variables or equations reducible to them proceed as :-

(i) Represent the unknown quantities by same variable $x$ and $y$, which are to be determined.

(ii) Find the conditions given in the problem and translate the verbal conditions into a pair of simultaneous linear equation.

(iii) Solve these equations & obtain the required quantities with appropriate units.

LINEAR EQUATIONS IN TWO VARIABLES II

Type of Problems :

(i) Determining two numbers when the relation between them is given,

(ii) Problems regarding fractions, digits of a number ages of persons.

(iii) Problems regarding current of a river, regarding time & distance.

(iv) Problems regarding menstruation and geometry.

(v) Problems regarding time & work

(vi) Problems regarding mixtures, cots of articles, porting & loss, discount et.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 5 Find two numbers such that the sum of twice the first and thrice the second is 89 and four times the first exceeds five times the second by 13 .

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the two numbers be $x$ and $y$.

Then, equation formed are $\quad 2 x+3 y=89$ …(i)

$4 x-5 y=13$ …(ii)

On solving eq. (i) & (ii) we get

$ \begin{aligned} & x=22 \\ & y=15 \end{aligned} $

Hence required numbers are $22$ & $15$.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 6 The numerator of a fraction is 4 less than the denominator If the numerator is decreased and the denominator is increased by 1 , then the denominator is eight time the numerator, find the reaction.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the numerator and denominator of a fraction be $x$ and $y$

Then, equation formed are $y-x=4$ …(i)

$ y+1=8(x-2) $ …(ii)

On solving eq. (i) & (ii) we get

$ x=3 $

and

$ y=7 \quad \text { Hence, fractions is } \frac{3}{7} \text {. } $

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 7 A number consists of two digits, the sum of the digits being 12. If 18 is subtracted from the number, the digits are reversed. Find the number

LINEAR EQUATIONS IN TWO VARIABLES II

Sol Let the two digits number be $1 y+x$

Then, equations formed are

$ 10 y+x-18=10 x+y \quad \Rightarrow \quad y-x=2$ …(i)
and $ \quad x+y=12$ …(ii)
On solving eq. (i) & (ii) we get
$ x=5 $
and $ y=7 $ Hence number is $ 75 .$

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 8 The sum of a two - digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3 , find the number

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let unit digit be $x$ ten’s digit be $y$ no. will be $10 y+x$.

Acc. to problem $(10 y+x)+(10 x+y)=165$

$\Rightarrow x+y=15$ …(i)

and $x-y=3$ …(ii)

or $-(x-y)=3$ …(iii)

On solving eq. (i) and (ii)

we gets $=9$ and $y=6 \quad \therefore \quad$ The number will be $69 . \quad$ Ans.

On solving eq. (i) and (iii)

we gets $x=6$ and $y=9 \quad \therefore \quad$ The number will be 96 . Ans.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 9 Six years hence a men’s age will be three times the age of his son and three years ago he was nine times as old as his son. Find their present ages

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let man’s present age be $x$ yrs & son’s present age be ’ $y$ ’ $yrs$.

According to problem $x+6=3(y+6)$ [After $6$ yrs]

and $x-3=9(y-3) \quad$ [Before $3$ yrs].

On solving equation (i) & (ii) we gets $x=30, y=6$.

So, the present age of $man=30$ years, present age of son $=6$ years.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 10 A boat goes $12 km$ upstream and $40 km$ downstream in $8 hrs$. It can go $16 km$. upstream and $32 km$ downstream in the same time. Find the speed of the boat it still water and the speed of the stream.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the speed of the boat in still water be $x km / hr$ and the speed of the stream be $ym / hr$ then speed of boat in downstream is $(x+y) km / hr$ and the speed of boat upstream is $(x-y) km / hr$.

Time taken to cover $12 km$ upstream $=\frac{12}{x-y}$ hrs.

Time taken to cover $40 km$ downstream $=\frac{40}{x+y} hrs$.

But, total time taken $8 hr$

LINEAR EQUATIONS IN TWO VARIABLES II

$\therefore \quad \frac{12}{x-y}+\frac{40}{x+y}=8$ …(i)

Time taken to cover $16 km$ upstream $=\frac{16}{x-y}$ hrs.

Time taken to cover $32 km$ downstream $=\frac{32}{x+y}$ hrs.

Total time taken $=8 hr$

$\therefore \quad \frac{16}{x-y}+\frac{32}{x+y}=8$ …(ii)

Solving equation (i) & (ii) we gets $x=6$ and $y=2$.

Hence, speed of boat in still water $=6 km / hr$ and speed of stream $=2 km / hr$.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 11 Ramesh travels $760 km$ to his home partly by train and partly by car. He taken $8 hr$, if he travels $160 km$ by train and the rest by car. He takes 12 minutes more, if he travels $240 km$ by train and the rest by car. Find the speed of train and the car.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the speed of train be $xm / hr$ & car be $y$ $km / hr$ respectively.

Acc. to problem $\frac{160}{x}+\frac{600}{y}=8$ …(i)

$\frac{240}{x}+\frac{520}{y}=\frac{41}{5}$ …(ii)

Solving equation (i) & (ii) we gets $x=80$ and $y=100$.

Hence, speed ot train $=80 km / hr$ and speed of car $=100 km / hr$.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 12 Points A and B are $90 km$ apart from each other on a highway. A car starts from A and another from B at the same time. If they go in the same direction, they meet in $9 hrs$ and if they go in opposite direction, they meet in $\frac{9}{7} hrs$. Find their speeds.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the speeds of the cars starting from A and B be $x km / hr$ and $ym / hr$ respectively.

Acc to problem $9 x-90=9 y$ …(i)

& $ \frac{9}{7} x+\frac{9}{7} y=90 $ …(ii)

Solving (i) & (ii) we gets $x=40$ & $y=30$.

Hence, speed of car starting from point $A=40 km / hr $ & speed of car starting from point $B=30 km / hr$.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 13 In a cyclic quadrilateral $ABCD, \angle A=(2 x+11)^{0}, \angle B=(y+12)^{0}, \angle C=(3 y+6)^{0}$ and $\angle D=(5 x-25)^{0}$, find the angles of the quadrilateral.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Acc. to problem $(2 x+11)^{0}+(3 y+6)^{0}=180^{0}$
$(y+12)^{0}+(5 x-25)^{0}=180^{0} $

Solving we get $ x=\frac{416}{13}$ & $ y=\frac{429}{13}$ $ \implies x=32 \text { and } y=33 $

$\therefore \quad \angle A=75^{\circ}, \angle B=45^{\circ}, \angle C=105^{\circ}, \angle D=135^{\circ}$

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 14 A vessel contains mixture of $24 \ell$ milk and $6 \ell$ water and a second vessel contains a mixture of $15 \ell$ milk & $10 \ell$ water. How much mixture of milk and water should be taken from the first and the second vessel separately and kept in a third vessel so that the third vessel may contain a mixture of $25 \ell$ milk and $10 \ell$ water?

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let $x \ell$ of mixture be taken from Ist vessel & $y \ell$ of the mixture be taken from $2^{\text {nd }}$ vessel and kept in 3rd vessel so that $(x+y) \ell$ of the mixture in third vessel may contain $25 \ell$ of milk & $10 \ell$ of water.

A mixture of $x \ell$ from 1 st vessel contains $\frac{24}{30} x=\frac{4}{5} x \ell$ of milk & $\frac{x}{5} \ell$ of water and a mixture of $y \ell$ from 2 nd vessel contains $\frac{3 y}{5} \ell$ of milk & $\frac{2 y}{5} \ell$ of water.

$\therefore \quad \frac{4}{5} x+\frac{3}{5} y=25$ …(i) $\quad$ $ \frac{x}{5}+\frac{2}{5} y=10 $ …(ii)

Solving (i) & (ii) $x=20$ litres and $y=15$ litres.

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 15 A lady has $25 p$ and $50 p$ coins in her purse. If in all she has 40 coins totaling Rs. 12.50, find the number of coins of each type she has.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let the lady has $x$ coins of $25 p$ and $y$ coins of $50 p$.

Then acc. to problem $\quad x+y=40$ …(i)

and $\quad 25 x+50 y=1250$ …(ii)

Solving for $x$ & $y$ we get $x=30$ (25 p coins) & $y=10$ (50 P coins).

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 16 Students of a class are made to stand in rows. If one student is extra in a row, there would be 2 rows less. If one students is less in row, there would be 3 rows more. Find the total number of students in the class.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let $x$ be the original no. of rows & $y$ be the original no. of student $s$ in each row.

$\therefore \quad$ Total no. of students $=xy$.

Acc. to problem

$(y+1)(x-2)=xy$ …(i)

and $(y-1)(x+3)=xy$ …(ii)

Solving (i) & (ii) to get

$x=12$ & $ y=5$ $\quad$ $\therefore \quad$ Total no. of students $=60$

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 17 A man started his job with a certain monthly salary and earned a fixed increment every year. If his salary was Rs. 4500 after 5 years. of service and Rs. 5550 after 12 years of service, what was his starting salary and what his annual increment.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let his initial monthly salary be Rs $x$ and annual increment be Rs $y$.

Then, Acc. to problem $\quad x+5 y=4500$ …(i)

$ x+12y=5550 $ …(ii)

Solving these two equations, we get $x=$ Rs. $3750 \ y=$ Rs 150 .

LINEAR EQUATIONS IN TWO VARIABLES II

Ex. 18 A dealer sold A VCR and a TV for Rs. 38560 making a profit of $12 %$ on CVR and 15% on TV. By selling them for Rs. 38620 , he would have realised a profit of $15 %$ on CVR and $12 %$ on TV. Find the cost price of each.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. Let C.P. of CVR be Rs x & C.P. of T.V. be Rs y.

Acc. to problem $\frac{112}{100} x+\frac{115}{100} y=38560$ …(i)

and

$ \frac{115}{100} \times+\frac{112}{100} y=38620 $ …(ii)

Solving for $x$ & $ y$ we get $x=$ Rs. 18000 & $y=$ Rs. 16000.

LINEAR EQUATIONS IN TWO VARIABLES II

DAILY PRACTIVE PROBLEMS 3

OBJECTIVE DPP 3.1

1. The graphs of $2 x+3 y-6=0,4 x-3 y-6=0, x=2$ and $y=\frac{2}{3}$ intersects in

(A) Four points $\quad$

(B) one point $\quad$

(C) two point $\quad$

(D) infinite number of points

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

2. The sum of two numbers is 20 , their product is 40 . The sum of their reciprocal is

(A) $\frac{1}{2}$ $\quad$

(B) 2 $\quad$

(C) 4 $\quad$

(D) $\frac{1}{10}$

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

3. If Rs. 50 is distributed among 150 children giving $50 p$ to each boy and $25 p$ to each girl. Then the number of boys is :

(A) 25 $\quad$

(B) 40 $\quad$

(C) 36 $\quad$

(D) 50

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

4. In covering a distance of $30 km$. Amit takes $2 hrs$. more than suresh. If Amit doubles his speed, he would take one hour less than suresh. Amits’ speed is :

(A) $5 km / hr$. $\quad$

(B) $7.5 km / hr$. $\quad$

(C) $6 km / hr$. $\quad$

(D) $6.2 km / hr$.

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

5. If in a fraction 1 less from two times of numerator & $1$ add in denominator then new fraction will be :

(A) $2(\frac{x-1}{y+1})$ $\quad$

(B) $\frac{2(x+1)}{y+1}$ $\quad$

(C) $(\frac{x}{y})$ $\quad$

(D) $\frac{2 x-1}{y+1}$

LINEAR EQUATIONS IN TWO VARIABLES II

LINEAR EQUATIONS IN TWO VARIABLES II

SUBJECTIVE DPP 3.2

1. The denominator of a fraction is greater than its numerator by 7. If 4 is added to both its numerator and denominator, then it becomes $\frac{1}{2}$. Find the fraction.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 1. $3 / 10$

LINEAR EQUATIONS IN TWO VARIABLES II

2. In a certain number is divided by the sum of its two digits, the quotient is 6 and remainder is 3 . If the digits are interchanged and the resulting number is divided by the sum of the digits, then the quotient is 4 and the remainder is 9 . Find the number.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 2. 75

LINEAR EQUATIONS IN TWO VARIABLES II

3. 2 men and 3 boys together can do a piece of work is 8 days. The same work si done in 6 days by 3 men and 2 boys together. How long would 1 boy alone or 1 man alone take to complete the work

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 3. One boy can do in 120 days and one man can do in 20 days.

LINEAR EQUATIONS IN TWO VARIABLES II

4. The um of two no $s$ is 18 . the sum of their reciprocal is $\frac{1}{4}$. Find the numbers.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 4. No. ’s are 12 and 6

LINEAR EQUATIONS IN TWO VARIABLES II

5. In a cyclic quadrilateral $ABCD, \angle A=(2 x+4)^{0}, \angle B=(y+3)^{0}, \angle C=(2 y+10)^{0}$ and $\angle D=(4 x-5)^{0}$ then find out the angles of quadrilateral.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 5. $A=70^{\circ}, B=53^{\circ}, C=110^{\circ}, D=127^{0}$

LINEAR EQUATIONS IN TWO VARIABLES II

6. Solve graphically and find the pints where the given liens meets the $y-$ axis : $2 x+y-11=0, x-y-1=0$.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 6. $x=4, y=3$ ,Point of contact with $x$ - axis $(0,11),(0,-1)$

LINEAR EQUATIONS IN TWO VARIABLES II

7. single graph paper & draw the graph of the following equations. Obtain the vertices of the triangles so obtained : $2 y-x=8,5 y-x=14$ & $y-2 x=1$.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 7. $(-4,2),(1,3),(2,5)$

LINEAR EQUATIONS IN TWO VARIABLES II

8. Draw the graph of $x-y+1=10 ; 3 x+2 y-12=0$. Calculate, the area bounded by these lines and $x-$ axis.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 8. $ 37.5$ Square units.

LINEAR EQUATIONS IN TWO VARIABLES II

9. A man sold a chair and a table together for Rs. 1520 thereby making a profit of $25 \%$ on chair and $10 \%$ on table. By selling them together for Rs. 1535 he would have made a profit of $10 \%$ on the chair and $25 \%$ on the table. Find cost price of each.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 9. $ Chair $=$ Rs. 600 , Tables $=$ Rs. 700$

LINEAR EQUATIONS IN TWO VARIABLES II

10. A man went to the Reserve Bank of India with a note or Rs. 500. He asked the cashier to give him Rs. 5 and Rs. 10 notes in return. The cashier gave him 70 notes in all. Find how many notes of Rs. 5 and Rs. 10 did the man receive.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 10. 5 rupees notes $=40$ & $10$ rupees notes $=30$

LINEAR EQUATIONS IN TWO VARIABLES II

11. Solve graphically: $5 x-6 y+30=0 ; 5 x+4 y-20=0$ Also find the vertices of the triangle formed by the above two lines and $x$-axis.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 11. $(0,5)$ vertices $(0,5)(-6,0),(4,0)$

LINEAR EQUATIONS IN TWO VARIABLES II

12. The sum of the digits of a two-digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18 . Find the number.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 12. 57

LINEAR EQUATIONS IN TWO VARIABLES II

13. Draw the graphs of the following equations and solve graphically:

$3 x+2 y+6=0 ; 3 x+8 y-12=0$

Also determine the co-ordinates of the vertices of the triangle formed by these lines and the $x$ - axis.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 13. $x=-4, y=3$, Lines meets $x$-axis at $(-2,0)$ & $(4,0)$

LINEAR EQUATIONS IN TWO VARIABLES II

14. A farmer wishes to purchase a number of sheep found the if they cost him Rs 42 a head, he would not have money enough by Rs 25; But if they cost him Rs 40 a head, he would them have Rs 40 more than he required; find the number of sheeps, and the money which he had.

LINEAR EQUATIONS IN TWO VARIABLES II

Sol. 14. 34 sheep, Rs 1400