knowledge-route Maths10 Ch3
title: “Lata knowledge-route-Class10-Math1-2 Merged.Pdf(1)” type: “reveal” weight: 1
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)
Ex. 1 Draw the graph of following :
(i)
(i)
LINEAR EQUATIONS IN TWO VARIABLES II
(ii)
LINEAR EQUATIONS IN TWO VARIABLES II
(iii)
LINEAR EQUATIONS IN TWO VARIABLES II
(iv)
Graphs of the type (ii) ay
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 2 Draw the graph of following : (i)
(i)
LINEAR EQUATIONS IN TWO VARIABLES II
(ii)
LINEAR EQUATIONS IN TWO VARIABLES II
(iii)
Graphs of the type (iii) ax + by
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 3 Draw the graph of following : (i)
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. (i)
1 | 4 | -3 | 0 | |
---|---|---|---|---|
1 | 4 | -3 | 0 |
(ii)
1 | -2 | 0 | |
---|---|---|---|
-1 | 2 | 0 |
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 :
LINEAR EQUATIONS IN TWO VARIABLES II
Sol.
(i)
0 | 1 | 2 | |
---|---|---|---|
-1 | 0 | 1 |
LINEAR EQUATIONS IN TWO VARIABLES II
(ii)
0 | 1 | 2 | |
---|---|---|---|
8 | 6 | 4 |
Solution is
Area of is
Area of
LINEAR EQUATIONS IN TWO VARIABLES II
LINEAR EQUATIONS IN TWO VARIABLES II
3.2 NATURE OF GRAPHICAL SOLUTION :
Let equations of two lines are
(i) Lines are consistent (unique solution) i.e. they meet at one point condition is
LINEAR EQUATIONS IN TWO VARIABLES II
(ii) Lines are inconsistent (no solution) i.e. they do not meet at one point condition is
LINEAR EQUATIONS IN TWO VARIABLES II
(iii) Lines are coincident (infinite solution) i.e. overlapping lines (or they are on one another) condition is
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
(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
Then, equation formed are
On solving eq. (i) & (ii) we get
Hence required numbers are
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
Then, equation formed are
On solving eq. (i) & (ii) we get
and
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
Then, equations formed are
and
On solving eq. (i) & (ii) we get
and
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
Acc. to problem
and
or
On solving eq. (i) and (ii)
we gets
On solving eq. (i) and (iii)
we gets
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
According to problem
and
On solving equation (i) & (ii) we gets
So, the present age of
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 10 A boat goes
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Let the speed of the boat in still water be
Time taken to cover
Time taken to cover
But, total time taken
LINEAR EQUATIONS IN TWO VARIABLES II
Time taken to cover
Time taken to cover
Total time taken
Solving equation (i) & (ii) we gets
Hence, speed of boat in still water
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 11 Ramesh travels
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Let the speed of train be
Acc. to problem
Solving equation (i) & (ii) we gets
Hence, speed ot train
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 12 Points A and B are
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Let the speeds of the cars starting from A and B be
Acc to problem
&
Solving (i) & (ii) we gets
Hence, speed of car starting from point
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 13 In a cyclic quadrilateral
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Acc. to problem
Solving we get
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 14 A vessel contains mixture of
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Let
A mixture of
Solving (i) & (ii)
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 15 A lady has
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. Let the lady has
Then acc. to problem
and
Solving for
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
Acc. to problem
and
Solving (i) & (ii) to get
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
Then, Acc. to problem
Solving these two equations, we get
LINEAR EQUATIONS IN TWO VARIABLES II
Ex. 18 A dealer sold A VCR and a TV for Rs. 38560 making a profit of
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
and
Solving for
LINEAR EQUATIONS IN TWO VARIABLES II
DAILY PRACTIVE PROBLEMS 3
OBJECTIVE DPP 3.1
1. The graphs of
(A) Four points
(B) one point
(C) two point
(D) infinite number of points
LINEAR EQUATIONS IN TWO VARIABLES II
Que. | 1 |
---|---|
Ans. | B |
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)
(B) 2
(C) 4
(D)
LINEAR EQUATIONS IN TWO VARIABLES II
Que. | 2 |
---|---|
Ans. | A |
LINEAR EQUATIONS IN TWO VARIABLES II
3. If Rs. 50 is distributed among 150 children giving
(A) 25
(B) 40
(C) 36
(D) 50
LINEAR EQUATIONS IN TWO VARIABLES II
Que. | 3 |
---|---|
Ans. | D |
LINEAR EQUATIONS IN TWO VARIABLES II
4. In covering a distance of
(A)
(B)
(C)
(D)
LINEAR EQUATIONS IN TWO VARIABLES II
Que. | 4 |
---|---|
Ans. | A |
LINEAR EQUATIONS IN TWO VARIABLES II
5. If in a fraction 1 less from two times of numerator &
(A)
(B)
(C)
(D)
LINEAR EQUATIONS IN TWO VARIABLES II
Que. | 5 |
---|---|
Ans. | D |
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
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 1.
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
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
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 5.
LINEAR EQUATIONS IN TWO VARIABLES II
6. Solve graphically and find the pints where the given liens meets the
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 6.
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 :
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 7.
LINEAR EQUATIONS IN TWO VARIABLES II
8. Draw the graph of
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 8.
LINEAR EQUATIONS IN TWO VARIABLES II
9. A man sold a chair and a table together for Rs. 1520 thereby making a profit of
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 9.
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
LINEAR EQUATIONS IN TWO VARIABLES II
11. Solve graphically:
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 11.
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:
Also determine the co-ordinates of the vertices of the triangle formed by these lines and the
LINEAR EQUATIONS IN TWO VARIABLES II
Sol. 13.
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