RD sharma class 10 Chapter 3 Linear Equations in Two Variables

Chapter 3 Linear Equations in Two Variables Download Pdf of RD sharma class 10 Chapter 3 Linear Equations in Two Variables

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Class 9 Linear Equations in two variables questions and answers

Questions on Linear Equations in two variables class 9

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Class 9 Chapter 4 (Linear Equations in Two Variables) Class Notes

 

Linear Equations in Two Variables

 

Linear Equations

The equation of a straight line is the linear equation. It could be in one variable or two variables.

Linear Equation in One Variable

The equation with one variable in it is known as a Linear Equation in One Variable.

The general form is px + q = s, where p, q and s are real numbers and p ≠ 0.

Example

x + 5 = 10

y – 3 = 19

These are called Linear Equations in One Variable because the highest degree of the variable is one.

Graph of the Linear Equation in One Variable

We can mark the point of the linear equation in one variable on the number line.

x = 2 can be marked on the number line as follows -

Linear Equation in Two Variables

An equation with two variables is known as a Linear Equation in Two Variables. The general form of the linear equation in two variables is

ax + by + c = 0

where a and b are coefficients and c is the constant. a ≠ 0 and b ≠ 0.

Example

6x + 2y + 5 = 0, etc.

Slope Intercept form

Generally, the linear equation in two variables is written in the slope-intercept form as this is the easiest way to find the slope of the straight line while drawing the graph of it.

The slope-intercept form is

Where m represents the slope of the line and b tells the point of intersection of the line with the y-axis.

Remark: If b = 0 i.e. if the equation is y = mx then the line will pass through the origin as the y-intercept is zero.

Solution of a Linear Equation

• There is only one solution in the linear equation in one variable but there are infinitely many solutions in the linear equation in two variables.

• As there are two variables, the solution will be in the form of an ordered pair, i.e. (x, y).

• The pair which satisfies the equation is the solution of that particular equation.

Example:

Find the solution for the equation 2x + y = 7.

Solution:

To calculate the solution of the given equation we will take x = 0

2(0) + y = 7

y = 7

Hence, one solution is (0, 7).

To find another solution we will take y = 0

2x + 0 = 7

x = 3.5

So another solution is (3.5, 0).

Graph of a Linear Equation in Two Variables

To draw the graph of linear equation in two variables, we need to draw a table to write the solutions of the given equation, and then plot them on the Cartesian plane.

By joining these coordinates, we get the line of that equation.

• The coordinates which satisfy the given Equation lies on the line of the equation.

• Every point (x, y) on the line is the solution x = a, y = b of the given Equation.

• Any point, which does not lie on the line AB, is not a solution of Equation.

Example:

Draw the graph of the equation 3x + 4y = 12.

Solution:

To draw the graph of the equation 3x + 4y = 12, we need to find the solutions of the equation.

Let x = 0

3(0) + 4y = 12

y = 3

Let y = 0

3x + 4(0) = 12

x = 4

Now draw a table to write the solutions.

x	0	4
y	3	0

Now we can draw the graph easily by plotting these points on the Cartesian plane.

Equations of Lines Parallel to the x-axis and y-axis

When we draw the graph of the linear equation in one variable then it will be a point on the number line.

x - 5 = 0

x = 5

This shows that it has only one solution i.e. x = 5, so it can be plotted on the number line.

But if we treat this equation as the linear equation in two variables then it will have infinitely many solutions and the graph will be a straight line.

x – 5 = 0 or x + (0) y – 5 = 0

This shows that this is the linear equation in two variables where the value of y is always zero. So the line will not touch the y-axis at any point.

x = 5, x = number, then the graph will be the vertical line parallel to the y-axis.

All the points on the line will be the solution of the given equation.

Similarly if y = - 3, y = number then the graph will be the horizontal line parallel to the x-axis.

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Linear Equations in two variables Quiz - 3

 Class 10 Linear Equations in Two Variables

Answer the questions:

(1) If thrice the daughter's age in years is added to mother's age, the sum is 62. If thrice the mother's age is added to the daughter's age, the sum is 122. Find the age of mother and daughter. 

(2) Two numbers are in ratio 2:1. If 4 is subtracted from both the numbers, ratio becomes 3:1. Find the numbers. 

(3) Which of the following conditions is true if the system of equations below is shown to be inconsistent \[\begin{array}{l}{a_1}x + {b_1}y + {c_1} = 0\\{a_2}x + {b_2}y + {c_2} = 0\end{array}\]

\[(a)\,\,\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}} \ne \frac{{{c_1}}}{{{c_2}}}\]

\[(b)\,\,\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}} \ne \frac{{{c_1}}}{{{c_2}}}\]

\[(c)\,\,\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} \ne \frac{{{c_1}}}{{{c_2}}}\]

\[(d)\,\,\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}\]

(4) 12 chairs and 10 tables cost Rs. 7800 and 4 chairs and 5 tables cost Rs. 3100. Find the cost of one chair and one table separately. 

(5) Sita is 6 years older than her friend Ashish. Sita's father is twice as old as Sita and Ashish is twice as old as his sister. The age of Sita's father is 39 years more than the age of Ashish's sister. Find the ages of Sita and Ashish. 

(6) Which of the following equations has a unique solution? 

   5x + 2y + 17 = 0, 10x + 4y + 14 = 0 

   5x + 2y + 8 = 0, 15x + 6y + 18 = 0 

   5x + 2y + 8 = 0, 10x + 4y + 16 = 0 

   5x + 2y + 8 = 0, 10x + 4y + 17 = 0 

(7) Four years ago George was four times older than his daughter. After four years, George will be 2 years more than two times the age of his daughter. Find the present age of George and his daughter. 

(8) Find value of k such that equations − x + 3y − 3 = 0 and − 3x + ky − 9 = 0, represents coincident lines. 

(9) It is given that the sum of digits of a two digit number is 13. If 27 is added to the number, the digits interchange their place. Find the number. 

(10) A two digit number is 6 more than 4 times the sum of its digits. If 27 is added to the number, the digit interchange their places. Find the number.

Choose correct answer(s) from the given choices 

(11) The pair of equations − 3x − y + 2 = 0 and − 6x − 4y + 6 = 0 have 

a. no solution

b. a unique solution

c. exactly two solutions

d. infinitely many solutions

(12) The pair of equations x = −3 and x = −8 has

a. no solution

b. infinitely mainy solution

c. two solutions

d. an unique solution

(13) For which value of p, following pair of linear equations have no solution.
p x + y = p² and  x + p y = 1 

a. 1

b. 0

c. −1

d. 0.5

(14) For which value of λ, following pair of linear equations have infinitely many solutions. 

λ x + 12 y = λ   and  3 x + λ y = λ − 3 

a. −3

b. 3

c. 6

d. -6

Answers: 

(1) 38 and 8 years 

(2) 16 and 8 

(3)$\frac{a_1}{a_2}=\frac{b_1}{b{}_2}\ne \frac{c_1}{c_2}$

(4) Rs. 400 and Rs. 300 

(5) 24 years and 18 years 

(6) 5x + 2y + 17 = 0, 10x + 4y + 14 = 0 

(7) 24 years and 9 years 

(8) 9 

(9) 58 

(10) 58 

(11) b. a unique solution 

(12) a. no solution 

(13) c. −1 

(14) c. 6 

Class 10th linear equations in two variables QUIZ

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Linear Equations in Two Variables Quiz 2

Class 10 (Linear Equations in two variables) Quiz 

Solve these linear equation in two variable ( x and y)

a. $37x+41y=70$
$41x+37y=86$

b. $99x+101y=499$
$101x+99y=501$

c. $23x-29y=98$
$29x-23y=110$

d. $ax+by=a-b$
$bx-ay=a+b$

e. $x+y=a+b$
$ax-by=a^2-b^2$

f. $(a-b)x+(a+b)y=a^2-2ab-b^2$
$(a+b)(x+y)=a^2-b^2$

g. $8x-3y=5xy$
$5y=-2xy$

h. $3(2x+y)=7xy$
$3(x+3y)=11xy$

i. $49x+51y=499$
$51x+49y=501$

j. $217x+131y=913$
$131x+217y=827$


Qustion 2
Solve these linear equation in two variable ( x and y)
i. $\frac{1}{2x}+\frac{1}{3y}=2$ 
$\frac{1}{3x}+\frac{1}{2y}=\frac{13}{6}$

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

iii. $\frac{22}{x+y}+\frac{15}{x-y}=5$
$\frac{55}{x+y}+\frac{45}{x-y}=14$

iv.$\frac{5}{x+y}-\frac{2}{x-y}=-1$
$\frac{15}{x+y}+\frac{7}{x-y}=10$

v. $bx+cy=a+b$
$ax(\frac{1}{a-b}-\frac{1}{a+b})+cy(\frac{1}{b-a}-\frac{1}{b+a})=\frac{2a}{a+b}$

vi) $\frac{1}{2(2x+3y)}+\frac{1}{7(3x-2y)}=\frac{17}{20}$
$\frac{7}{(2x+3y)}-\frac{1}{(3x-2y)}=-\frac{28}{5}$

vii.$\frac{x+1}{2}-\frac{y+4}{11}=2$
$\frac{x+3}{2}+\frac{2y+3}{17}=5$
viii.$\frac{7x-2y}{xy}=5$
$\frac{8x+7y}{xy}=15$

ix. $\frac{x}{a}+\frac{y}{b}=2$
$ax-by=a^2-b^2$

x.$\frac{57}{x+y}+\frac{6}{x-y}=5$
$\frac{38}{x+y}+\frac{21}{x-y}=9$

Answer

1.
i. (3, 2)
ii. (3, -1)
iii. (8, 3)
iv. (1,-1)
v. (a, b)
vi. (a,-b)
vii. (0, 0) ,(22/31, 11/23)
viii. (0,0) (1,3/2)
ix. (1 ½ , 9/2)
x. 3,2

Q2.
i. (1/2, 1/3)
Hint: Take $\frac{1}{6x}=p,\frac{1}{6y}=q$ and then solve in p& q and then find x and y

ii. (1 ,3)
Hint: Take $\frac{1}{x}=p,\frac{1}{y}=q$ and then solve in p& q and then find x and y

iii. (8,3)
Hint: Take $\frac{1}{x+y}=p,\frac{1}{x-y}=q$ and then solve in p& q and then find x and y

iv. (3,2)
Hint: Take $\frac{1}{x+y}=p,\frac{1}{x-y}=q$ and then solve in p& q and then find x and y

v. $\frac{a}{b},\frac{b}{c}$ vi. (2,1)
Hint: Take $\frac{1}{2x+3y}=p,\frac{1}{3x-2y}=q$ and then solve in p& q and then find x and y

vii. 5, 7
viii. (1, 1)
Hint:
Convert into these forms
$\frac{7}{y}-\frac{2}{x}=5$
$\frac{8}{y}+\frac{7}{y}=15$
Take $\frac{1}{x}=p,\frac{1}{y}=q$ and then solve in p& q and then find x and y

ix. (a,b)
x. (11, 8)

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