Class 10 (All in one) – Pair of Linear Equations in Two variablesContact Number: 9667591930 / 8527521718

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Five years ago, Jacob’s age was seven times that of his son. After five years. The age of Jacob will be three times that of his son. Represent this situation algebraically and graphically.

Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions.

$15x-30y+1=0,3x-\frac{24}{4}y+\frac{1}{5}=0$

Check graphically, whether the following pair of linear equations is consistent. If yes, solve it graphically.

2*x* – *y* = 0, *x* + *y* = 0

Solve graphically, the pair of linear equations *x –* *y* = –1 and 2*x* + *y* – 10 = 0. Also, find the vertices of the triangle formed by these lines and *X-*axis.

Solve graphically the pair of linear equations 3*x* + *y* – 11 = 0, *x* – *y* – 1 = 0. Also, find the vertices of the triangle formed by these lines and *Y*-axis.

Two straight paths are represented by the lines 7*x* – 5*y* = 3 and 21*x* – 15*y* = 5. Check whether the paths cross each other.

Write a pair of linear equations which has the unique solution *x* = –1 and *y* = 3. How many such pairs can you write? **NCERT Exemplar **

On comparing the ratios $\frac{{a}_{1}}{{a}_{2}},\frac{{b}_{1}}{{b}_{2}}$ and $\frac{{c}_{1}}{{c}_{2}}$, find out whether the following pairs of linear equations are consistent or inconsistent?

(i) $\frac{4}{3}x+2y=8$ and $2x+3y=12$

(ii) 4*x* – *y* = 4 and 3*x* + 2*y* = 14

(iii) 3*x* – 5*y* = 11 and 6*x* – 10*y* = 7

(iv) 6*x –* 3*y* = 12 and 2*x* – *y* = 4

4 chairs and 3 tables cost ₹ 210 and 5 chairs and 2 tables cost ₹ 175. Find the cost of one chair and table separately.

Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio becomes 4:5. Find the numbers. **NCERT Exemplar**

Find the solution of the following system of equation by substitution method.

(i) *x* + *y* = 8, 2*x* – 3*y* = 1

(ii) 3*x* + 2*y* = 10, 12*x* + 8*y* = 30

(iii) 2*x* – 7*y* = 11, 6*x* – 21*y* = 33

(iv) $\sqrt{2}x+\sqrt{5}y=0,\sqrt{6}x+\sqrt{15}y=0$

Solve the following pair of linear equations by substitution method.

(i) *x* – *y* = 2, 3*x* + 2*y* = 16

(ii) 7*x* - 4*y* = 3, *x* + 2*y* = 3

(iii) 3*x* + 7*y* = 37, 5*x* + 6*y* = 39

(iv) $\frac{3x-4y}{2}=10,\frac{3x+2y}{4}=2$

(v) $y=\frac{2}{3}x+6,2y-4x=20$

(vi) $3x-\frac{y+7}{11}=8,2y+\frac{x+11}{7}=10$

(vii) 11*x* + 15*y* + 23 = 0, 0.7*x* – 0.2*y* = 2

(viii) $\sqrt{7}x+\sqrt{11}y=0,\sqrt{3}x-\sqrt{5}y=0$

(i) Solve 2*x* – 3*y* = 13 and 7*x* – 2*y* = 20 and hence find the value of *m* for which *y* = *mx* + 7.

(ii) Solve 5*x* + 4*y* = 10 and 3*x* – 2*y* = 0 and hence find the value of *m* for which *y* = *mx* + 3.

Solve for *x *and *y* by substitution method

(i) *x* + *y* = *a* – b, *ax *– *by* = *a*^{2} + *b*^{2}

(ii) $\frac{x}{a}+\frac{y}{b}=2,ax\u2013by={a}^{2}-{b}^{2}$

(iii) $\frac{bx}{a}-\frac{ay}{b}+a+b=0,bx\u2013ay+2ab=0$

(iv) 2(*ax* – *by*) + *a* + 4*b* = 0, 2(*bx* + *ay*) + (*b* – 4*a*) = 0

(v) $2\left(\frac{x}{a}\right)+\frac{y}{b}=2,\frac{x}{a}-\frac{y}{b}=4$

A man has ₹ 100 in ₹ 1 coins and 50 paise coins. All the 50 paise coins are worth as much as all the ₹ 1 coins. How many coins of each he has?

In $\Delta ABC,\angle C=5\angle B=3(\angle A+\angle B)$, find all angles of $\Delta ABC$.

The perimeter of rectangular lawn is 54 m. It is reduced in size, so than length is $\frac{3}{5}\text{th}$ and breadth is $\frac{3}{4}\text{th}$ of the original dimensions. The perimeter of the reduced rectangle is 36 m. What were the original dimensions of the lawn?

Solve the following pair of equations by using elimination method.

(i) 8*x* + 5*y* = 11, *x* + *y* = 4

(ii) *x* – *y* = 3, 3*x* – 2*y* = 10

(iii) 4*x* – 3*y* = 8, 6*x* – *y* = $\frac{29}{3}$

(iv) 11*x* + 15*y* + 23 = 0, 7*x* – 2*y* – 20 = 0

(v) 3*x* – 5*y* = 4, 9*x *– 2*y* – 7 = 0

(vi) *x* – 2*y* = 3, 3*x* – 3*y* = 5

(vii) 0.4*x* + 0.3*y* =1.7, 0.7*x* – 0.2*y* = 0.8

(viii) $\frac{x}{4}+\frac{y}{3}=-\frac{1}{12},\frac{x}{2}-\frac{5}{4}y=\frac{7}{4}$

(ix) $\frac{x}{3}+\frac{y}{3}=14,\frac{5x}{6}-\frac{y}{3}+7=0$

Solve the following pair of linear equations by elimination method.

(i) *ax* + *by* = *c*, *a*^{2}*x* + *b*^{2}*y* = *c*^{2}

(ii) (*a* + *b*) *x* + (*a* – *b*) *y* = 2*ab*, (*a* + *b*) *x* – (*a* – *b*) *y* = *ab*^{(iii) }$\frac{bx}{a}+\frac{ay}{b}={a}^{2}+{b}^{2},x+y=2ab$

Find the solution of the pair of equations $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{x}{8}+\frac{y}{6}=15$ and find λ if *y* = λ*x* + 5. **NCERT Exemplar**

The sum of the digits of a two-digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number. **CBSE 2015**

The area of a rectangle gets reduced by 80 sq. units, if its length is reduced by 5 units and the breadth is increased by 2 units. If we increase the length by 10 units and decrease the breadth by 5 units, then the area is increased by 50 sq. units. Find the length and the breadth of the rectangle. **CBSE 2015**

Sheena went to a bank to withdraw ₹1000 and asked the cashier to give her ₹ 100 and ₹ 50 notes only. She got 14 notes in all. Find how many notes ₹ 100 and ₹ 50 she received?

Solve the following pair of linear equations by cross-multiplication method.

(i) 4*x* + 6*y* = 5, 2*x* + 9*y* = 3

(ii) 5*x* + 4*y* – 4 = 0, *x* – 12*y* – 20 = 0

(iii) 3*x* – 5*y* + 1 = 0, 2*x* + 3*y* – 12 = 0

(iv) $\frac{x}{7}+\frac{y}{3}=5,\frac{x}{2}-\frac{y}{9}=6$

Solve the following pair of linear equations by cross-multiplication method.

(i) *ax* + *by* = *a*^{2}, *bx* – *ay* = *ab*

(ii) (*a *– *b*)* x* + (*a* + *b*)*y* = 2*a*^{2} – 2*b*^{2}, *ax* – *by* = *a*^{2} + *b*^{2}

(iii) $\frac{x}{a}+\frac{y}{b}=a+b,\frac{x}{{a}^{2}}-\frac{y}{{b}^{2}}=2$

(iv) $ax+by=1,bx+ay=\frac{2ab}{{a}^{2}+{b}^{2}}$

Find the value of *k* for which the following system of equations has a unique solution.

(i) *kx* + 2*y* = 5, 3*x* + *y* = 1

(ii) *x* – 3*y* = 5, 5*x* + *ky* = 10

(iii) *kx* + 2*y* = 4, 8*x* + *ky* = 11

(iv) 4*x* – 5*y* = *k*, 2*x* – 3*y* = 12

Find the value of *k* for which the following system of equations has no solution.

(i) 4*x* + 5*y* = 17, *kx* + 15*y* = 33

(ii) *kx* + 2*y* = 5, 8*x* + *ky* = 20

(iii) 3*x* + *y* = 1, (2*k* – 1)*x* + (*k *– 1)*y* = (2*k* + 1)

(iv) 3*x* – *y* – 5 = 0, 6*x* – 2*y* + *k* = 0 **CBSE 2008**

Find the value of *k* for which the following system of equations has infinitely many solutions.

(i) 4*x* + 7*y* = 10; (*k* + 2)*x* + 21*y* = 3*k*

(ii) 2*x* + (*k* – 2)*y* = *k*; 6*x* + (2*k* – 1)*y* = 2*k* + 5

(iii) 2*x* + 3*y* = 2; (*k* + 2)*x* + (2*k *+ 1) = 2(*k* – 1)

(iv) *x* + (*k* + 1)*y* = 5

(*k* + 1)*x* + 9*y* = (8*k *– 1)

Find the values of p and q for which the following system of equations has infinitely many solutions.

(i) (2*p* – 1)*x* + 3*y* = 5

3*x* + (*q* – 1)*y* = 2

(ii) 2*x* + 3*y* = 7;

(*p* + *q* + 1)*x* + (*p* + 2*q* + 2)*y* = 4(*p* + *q*) + 1

Find the value of *k*, for which system of equations *kx* + 3*y* = 3 and 12*x* + *ky* = 6 represent parallel lines.

For what value of *k*, the pair of linear equations *x* + 2*y* = 3, 5*x* + *ky* + 7 = 0 represents

(i) Intersecting lines

(ii) Parallel lines

Is there any value of *k* for which the given equations represents coincident lines?

HOTS solve the following pair of linear equations by cross-multiplication method.

(i) 2(*ax* – *by*) + *a* + 4*b* = 0

2(*bx* + *ay*) + *b* – 4*a* = 0

(ii) (*a* + 2*b*)*x* + (2*a* – *b*)*y* = 2

(*a* – 2*b*)*x* + (2*a* + *b*)*y* = 3

(iii) (*a* – *b*)*x* + (*a* + *b*)*y* = 2*a*^{2} – 2*b*^{2};

(*a* + *b*) (*x* + *y*) = 4*ab*

Solved the following pair of equations of reducing them into a pair of linear equations.

(i) $\frac{2}{x}+\frac{3}{y}=22,\frac{5}{x}-\frac{4}{y}=9$

(ii) $\frac{5}{x}+6y=13,\frac{3}{x}+4y=7$

(iii) $x+\frac{6}{y}=6,3x-\frac{8}{y}=5$

(iv) 4*x* + 6*y* = 3*xy*, 8*x* + 9*y* = 5*xy*

(v) $\frac{2}{x}+\frac{3}{y}=\frac{9}{xy},\frac{4}{x}+\frac{9}{y}=\frac{21}{xy},x\ne 0,y\ne 0$

(vi) $\frac{3y-x}{xy}=-9,\frac{2y+3x}{xy}=5,xy\ne 0$

(vii) $\frac{5}{x+1}-\frac{2}{y-1}=\frac{1}{2},\frac{10}{x+1}+\frac{2}{y-1}=\frac{5}{2},x\ne -1,y\ne 1$

(viii) $\frac{6}{x+y}=\frac{7}{x-y}+3,\frac{1}{2(x+y)}=\frac{1}{3(x-y)},x+y\ne 0,x-y\ne 0$

A man travels 600 km partly by train and partly by car. It takes 8 h and 40 min. if he travels 320 km by train and the rest by car. It would take 30 min. more, if he travels 200 km by train and the rest by car. Find the speed of the train and the car separately. **CBSE 2011**

If a motorboat can travel 30 km upstream and 28 km downstream in 7 h, it can travel 21 km upstream and return in 5 h. Find the speed of the boat in still water and the speed of the stream. **NCERT Exemplar**

8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken to finish the work by one man alone.

Reduce the following pair of equations into a pair of linear equations and solve them. **NCERT Exemplar**

(i) $\frac{2xy}{x+y}=\frac{3}{2},\frac{xy}{2x-y}=\frac{-3}{10};x+y\ne 0,2x-y\ne 0$

(ii) $\frac{2}{3x+2y}+\frac{3}{3x-2y}=\frac{17}{5},\frac{5}{3x+2y}+\frac{1}{3x-2y}=2$

Students were standing in rows for a mass drill. If one student is extra in a row, there would be 2 rows less. If one students is less in a row, there would be 3 rows more. Find the number of students.

It can take 12 h to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 h and the pipe of smaller diameter for 9 h. only half the pool can be filled. How long would it take for each pipe to fill the pool separately?

Find the solution of pair of equations *y* = 0 and *y* = –6.

If *x* = *a*, *y* = *b* is the solution of the equations *x* – *y* = 2 and *x* + *y* = 4, then find the values of *a* and *b*. **NCERT Exemplar**

Do the equations *x* + 3*y* – 1 = 3 and 2*x* + 6*y* = 6 represent a pair of coincident lines? Justify your answer.

For which values of *c*, the pair of equations 2*x* + 2*y* = 8 and 8*x* + 10*y* = *c* have a unique solution?

Obtain the condition for the following pair of linear equations to have a unique solution.*ax* + *by* = *c* and *lx* + *my* = *n*

What should be the value of λ, for the given equations to have infinitely many solutions?

5*x* + λ*y* = 4 and 15*x* + 3*y* = 12

For which value of *p*, the pair of equations 6*x* + 5*y* = 4 and 12*x* + *py* = – 8 has no solution?

Find the value of *k*, so that the following system of equations has no solution.

3*x* – *y* – 5 = 0, 6*x* – 2*y* – *k* = 0

Find the value of *x* + *y*, if 3*x* – 2*y* = 5 and 3*y* – 2*x* = 3.

HOTS in the given figure, *ABCD* is parallelogram. Find the values of *x* and *y*.

Determine the values of *a* and *b*, for which the following pairs of linear equations has infinitely many solutions.

3*x* – (*a* + 1) *y* = 2*b* – 1

and 5*x* + (1 – 2*a*) *y* = 3*b*

Two straight paths are represented by the lines 7*x* – 5*y* = 3 and 14*x* – 10*y* = 5. Check whether the paths cross each other.

Father’s age is 3 times the sum of ages of his two children. After 5yrs, his age will be twice the sum of ages of the two children. Find the age of father. **CBSE 2011, 10**

Find the values of *x* and *y* in the given rectangle. **NCERT Exemplar**

Find a, if the line 3*x* + *ay* = 8 passes through the intersection of lines represented by equations 3*x* – 2*y* = 10 and 5*x* + *y* = 8.

There are some students in the two examination halls *A* and *B*. To make the number of students equal in each hall, 10 students are sent from *A* and *B*.

But if 20 students are sent from *B* to *A*, the number of students in *A* becomes double the number of students in B.

Find the number of students in the two halls. **NCERT Exemplar**

If the angles of a triangle are *x*, *y* and $40\xb0$ and the difference between the two angles *x* and *y* is $30\xb0$.Then, find the values of *x* and *y*. **NCERT Exemplar**

For which values(s) of λ, does the pair of linear equations λ*x* + *y* = λ^{2 }and *x* + λ*y* = 1 have **NCERT Exemplar**

(i) no solution?

(ii) infinitely many solutions?

(iii) a unique solutions?

Find the values of *a* and *b* for which the following system of linear equations has infinite number of solutions. **CBSE 2012**

(*a *+ *b*)*x *– 2*by* = 5*a *+ 2*b* + 1

and 3*x – y* = 14

A sailor goes 8 km downstream in 40 min and comes back in 1h. Find the speed of sailor in still water and the speed of current. **CBSE 2010**

Determine algebraically, the vertices of the triangle formed by the lines

3*x* ‒ *y* = 3, 2*x *‒ 3*y* = 2 and *x* + 2*y* = 8. **NCERT Exemplar**

Solve graphically, the pair of equations 2*x* + *y* = 6 and 2*x* ‒ *y* + 2 = 0. Find the ratio of the areas of the two triangles formed by the lines representing these equations with *X*-axis and the lines with *Y*-axis.

*A* and *B *each have a certain number of mangoes. *A* says to *B*, if you give 30 of your mangoes, I will have twice as many as left with you, *B* replies, if you give me 10, I will have thrice as many as left with you. How many mangoes does each have?

The sum of a two-digit number and number obtained by reversing the order of digits is 99. If the digits of the number differ by 3, then find the numbers. **CBSE2010**

Find the point of intersection of lines

2*ax* ‒ *by* = 2*a*^{2} ‒ *b*^{2}

and *ax* + 2*by* = *a*^{2} + 2*b*^{2}

by eliminating the variables. Show that the system of equations is concurrent with the line represented by equation

(*a* ‒ *b*)*x* + (*a* + *b*)*y* = *a*^{2} + *b*^{2}

Solve the system of following equations

$\frac{1}{2(2x+3y)}+\frac{12}{7(3x-2y)}=\frac{1}{2}$

and $\frac{7}{2x+3y}+\frac{4}{3x-2y}=2$

Solve the following equations for *x* and *y*.

7* ^{x}* + 5

7

A railway half ticket cost half the full fare but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the stations *A* to *B* costs ₹ 2530. Also, one reserved first class ticket and one reserved first class half ticket from stations *A* to *B* costs ₹3810. Find the full first class fare from stations *A* to *B* and also the reservation charges for a ticket. **NCERT Exampler**

The Resident Welfare Association of a colony decided to build two straight paths in their neighbourhood park such that they do not cross each other, to plant trees along the boundary lines of each path.

One of the members of association, Sarika suggested that the paths should be constructed represented by the two linear equations *x* ‒ 3*y* = 2 and ‒2*x* + 6*y* = 5. Check whether the two paths will cross each other or not. What value is depicted from this action?

While teaching about the Indian National flag, teacher asked the students that how many lines are there in blue colour wheel? One student replies that it is 8 times the number of colours in the flag. While other says that the sum of the number of colours in the flag and number of lines in the wheel of the flag is 27. Convert the statements given by the students into linear equation of two variables. Find the number of lines in the wheel. What does the wheel signifies in the flag?

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