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Page 1
1
· Methods of solving linear equations in two variables - graphical method,
Cramer’s method
· Equations that can be transformed in linear equation in two variables
· Application of simultaneous equations
Let’s recall.
Linear equation in two variables
An equation which contains two variables and the degree of each term
containing variable is one, is called a linear equation in two variables.
ax + by + c = 0 is the general form of a linear equation in two variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Ex. 3x - 4y + 12 = 0 is the general form of equation 3x = 4y - 12
Activity : Complete the following table
No. Equation Is the equation a linear equation in 2
variables ?
1
4m + 3n = 12
Yes
2
3x
2
- 7y = 13
3
2 x - 5 y = 16
4
0x + 6y - 3 = 0
5
0.3x + 0y -36 = 0
6
45
4
xy
7
4xy - 5y - 8 = 0
1
Linear Equations in Two Variables
Let’s study.
Page 2
1
· Methods of solving linear equations in two variables - graphical method,
Cramer’s method
· Equations that can be transformed in linear equation in two variables
· Application of simultaneous equations
Let’s recall.
Linear equation in two variables
An equation which contains two variables and the degree of each term
containing variable is one, is called a linear equation in two variables.
ax + by + c = 0 is the general form of a linear equation in two variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Ex. 3x - 4y + 12 = 0 is the general form of equation 3x = 4y - 12
Activity : Complete the following table
No. Equation Is the equation a linear equation in 2
variables ?
1
4m + 3n = 12
Yes
2
3x
2
- 7y = 13
3
2 x - 5 y = 16
4
0x + 6y - 3 = 0
5
0.3x + 0y -36 = 0
6
45
4
xy
7
4xy - 5y - 8 = 0
1
Linear Equations in Two Variables
Let’s study.
2
Simultaneous linear equations
When we think about two linear equations in two variables at the same
time, they are called simultaneous equations.
Last year we learnt to solve simultaneous equations by eliminating one
variable. Let us revise it.
Ex. (1) Solve the following simultaneous equations.
(1) 5x - 3y = 8; 3x + y = 2
Solution :
Method I : 5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Multiplying both sides of
equation (II) by 3.
9x + 3y = 6 . . . (III)
5x - 3y = 8. . . (I)
Now let us add equations (I)
and (III)
5x - 3y = 8
9x + 3y = 6
14x = 14
\ x = 1
substituting x = 1 in equation (II)
3x + y = 2
\ 3 ´ 1 + y = 2
\ 3 + y = 2
\ y = -1
solution is x = 1, y = -1; it is also
written as (x, y) = (1, -1)
Method (II)
5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Let us write value of y in terms
of x from equation (II) as
y = 2 - 3x . . . (III)
Substituting this value of y in
equation (I).
5x - 3y = 8
\ 5x - 3(2 - 3x) = 8
\ 5x - 6 + 9x = 8
\ 14x - 6 = 8
\ 14x = 8 + 6
\ 14x = 14
\ x = 1
Substituting x = 1 in equation
(III).
y = 2 - 3x
\ y = 2 - 3 ´ 1
\ y = 2 - 3
\ y = -1
x = 1, y = -1 is the solution.
+
Page 3
1
· Methods of solving linear equations in two variables - graphical method,
Cramer’s method
· Equations that can be transformed in linear equation in two variables
· Application of simultaneous equations
Let’s recall.
Linear equation in two variables
An equation which contains two variables and the degree of each term
containing variable is one, is called a linear equation in two variables.
ax + by + c = 0 is the general form of a linear equation in two variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Ex. 3x - 4y + 12 = 0 is the general form of equation 3x = 4y - 12
Activity : Complete the following table
No. Equation Is the equation a linear equation in 2
variables ?
1
4m + 3n = 12
Yes
2
3x
2
- 7y = 13
3
2 x - 5 y = 16
4
0x + 6y - 3 = 0
5
0.3x + 0y -36 = 0
6
45
4
xy
7
4xy - 5y - 8 = 0
1
Linear Equations in Two Variables
Let’s study.
2
Simultaneous linear equations
When we think about two linear equations in two variables at the same
time, they are called simultaneous equations.
Last year we learnt to solve simultaneous equations by eliminating one
variable. Let us revise it.
Ex. (1) Solve the following simultaneous equations.
(1) 5x - 3y = 8; 3x + y = 2
Solution :
Method I : 5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Multiplying both sides of
equation (II) by 3.
9x + 3y = 6 . . . (III)
5x - 3y = 8. . . (I)
Now let us add equations (I)
and (III)
5x - 3y = 8
9x + 3y = 6
14x = 14
\ x = 1
substituting x = 1 in equation (II)
3x + y = 2
\ 3 ´ 1 + y = 2
\ 3 + y = 2
\ y = -1
solution is x = 1, y = -1; it is also
written as (x, y) = (1, -1)
Method (II)
5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Let us write value of y in terms
of x from equation (II) as
y = 2 - 3x . . . (III)
Substituting this value of y in
equation (I).
5x - 3y = 8
\ 5x - 3(2 - 3x) = 8
\ 5x - 6 + 9x = 8
\ 14x - 6 = 8
\ 14x = 8 + 6
\ 14x = 14
\ x = 1
Substituting x = 1 in equation
(III).
y = 2 - 3x
\ y = 2 - 3 ´ 1
\ y = 2 - 3
\ y = -1
x = 1, y = -1 is the solution.
+
3
Ex. (2) Solve : 3x + 2y = 29; 5x - y = 18
Solution : 3x + 2y = 29. . . (I) and 5x - y = 18 . . . (II)
Let’s solve the equations by eliminating ’y’. Fill suitably the boxes below.
Multiplying equation (II) by 2.
\ 5x ´ - y ´ = 18 ´
\ 10x - 2y = . . . (III)
Let’s add equations (I) and (III)
3x + 2y = 29
+
- =
= \ x =
Substituting x = 5 in equation (I)
3x + 2y = 29
\ 3 ´ + 2y = 29
\ + 2y = 29
\ 2y = 29 -
\ 2y = \ y =
(x, y) = ( , ) is the solution.
Ex. (3) Solve : 15x + 17y = 21; 17x + 15y = 11
Solution : 15x + 17y = 21. . . (I)
17x + 15y = 11 . . . (II)
In the two equations above, the coefficients of x and y are interchanged.
While solving such equations we get two simple equations by adding and
subtracting the given equations. After solving these equations, we can easily find
the solution.
Let’s add the two given equations.
15x + 17y = 21
17x + 15y = 11
32x + 32y = 32
+
Page 4
1
· Methods of solving linear equations in two variables - graphical method,
Cramer’s method
· Equations that can be transformed in linear equation in two variables
· Application of simultaneous equations
Let’s recall.
Linear equation in two variables
An equation which contains two variables and the degree of each term
containing variable is one, is called a linear equation in two variables.
ax + by + c = 0 is the general form of a linear equation in two variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Ex. 3x - 4y + 12 = 0 is the general form of equation 3x = 4y - 12
Activity : Complete the following table
No. Equation Is the equation a linear equation in 2
variables ?
1
4m + 3n = 12
Yes
2
3x
2
- 7y = 13
3
2 x - 5 y = 16
4
0x + 6y - 3 = 0
5
0.3x + 0y -36 = 0
6
45
4
xy
7
4xy - 5y - 8 = 0
1
Linear Equations in Two Variables
Let’s study.
2
Simultaneous linear equations
When we think about two linear equations in two variables at the same
time, they are called simultaneous equations.
Last year we learnt to solve simultaneous equations by eliminating one
variable. Let us revise it.
Ex. (1) Solve the following simultaneous equations.
(1) 5x - 3y = 8; 3x + y = 2
Solution :
Method I : 5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Multiplying both sides of
equation (II) by 3.
9x + 3y = 6 . . . (III)
5x - 3y = 8. . . (I)
Now let us add equations (I)
and (III)
5x - 3y = 8
9x + 3y = 6
14x = 14
\ x = 1
substituting x = 1 in equation (II)
3x + y = 2
\ 3 ´ 1 + y = 2
\ 3 + y = 2
\ y = -1
solution is x = 1, y = -1; it is also
written as (x, y) = (1, -1)
Method (II)
5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Let us write value of y in terms
of x from equation (II) as
y = 2 - 3x . . . (III)
Substituting this value of y in
equation (I).
5x - 3y = 8
\ 5x - 3(2 - 3x) = 8
\ 5x - 6 + 9x = 8
\ 14x - 6 = 8
\ 14x = 8 + 6
\ 14x = 14
\ x = 1
Substituting x = 1 in equation
(III).
y = 2 - 3x
\ y = 2 - 3 ´ 1
\ y = 2 - 3
\ y = -1
x = 1, y = -1 is the solution.
+
3
Ex. (2) Solve : 3x + 2y = 29; 5x - y = 18
Solution : 3x + 2y = 29. . . (I) and 5x - y = 18 . . . (II)
Let’s solve the equations by eliminating ’y’. Fill suitably the boxes below.
Multiplying equation (II) by 2.
\ 5x ´ - y ´ = 18 ´
\ 10x - 2y = . . . (III)
Let’s add equations (I) and (III)
3x + 2y = 29
+
- =
= \ x =
Substituting x = 5 in equation (I)
3x + 2y = 29
\ 3 ´ + 2y = 29
\ + 2y = 29
\ 2y = 29 -
\ 2y = \ y =
(x, y) = ( , ) is the solution.
Ex. (3) Solve : 15x + 17y = 21; 17x + 15y = 11
Solution : 15x + 17y = 21. . . (I)
17x + 15y = 11 . . . (II)
In the two equations above, the coefficients of x and y are interchanged.
While solving such equations we get two simple equations by adding and
subtracting the given equations. After solving these equations, we can easily find
the solution.
Let’s add the two given equations.
15x + 17y = 21
17x + 15y = 11
32x + 32y = 32
+
4
Dividing both sides of the equation by 32.
x + y = 1 . . . (III)
Now, let’s subtract equation (II) from (I)
15x + 17y = 21
17x + 15y = 11
-2x + 2y = 10
dividing the equation by 2.
-x + y = 5 . . . (IV)
Now let’s add equations (III) and (V).
x + y = 1
-x + y = 5
\ 2y = 6 \ y = 3
Place this value in equation (III).
x + y = 1
\ x + 3 = 1
\ x = 1 - 3 \ x = -2
(x, y) = (-2, 3) is the solution.
- - -
Practice Set 1.1
(1) Complete the following activity to solve the simultaneous equations.
5x + 3y = 9 -----(I)
-
+
+
2x - 3y = 12 ----- (II)
Let’s add eqations (I) and (II).
5x + 3y = 9
2x - 3y = 12
x =
x = x =
Place x = 3 in equation (I).
5 ´ + 3y = 9
3y = 9 -
3y =
y =
3
y =
\ Solution is (x, y) = ( , ).
Page 5
1
· Methods of solving linear equations in two variables - graphical method,
Cramer’s method
· Equations that can be transformed in linear equation in two variables
· Application of simultaneous equations
Let’s recall.
Linear equation in two variables
An equation which contains two variables and the degree of each term
containing variable is one, is called a linear equation in two variables.
ax + by + c = 0 is the general form of a linear equation in two variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Ex. 3x - 4y + 12 = 0 is the general form of equation 3x = 4y - 12
Activity : Complete the following table
No. Equation Is the equation a linear equation in 2
variables ?
1
4m + 3n = 12
Yes
2
3x
2
- 7y = 13
3
2 x - 5 y = 16
4
0x + 6y - 3 = 0
5
0.3x + 0y -36 = 0
6
45
4
xy
7
4xy - 5y - 8 = 0
1
Linear Equations in Two Variables
Let’s study.
2
Simultaneous linear equations
When we think about two linear equations in two variables at the same
time, they are called simultaneous equations.
Last year we learnt to solve simultaneous equations by eliminating one
variable. Let us revise it.
Ex. (1) Solve the following simultaneous equations.
(1) 5x - 3y = 8; 3x + y = 2
Solution :
Method I : 5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Multiplying both sides of
equation (II) by 3.
9x + 3y = 6 . . . (III)
5x - 3y = 8. . . (I)
Now let us add equations (I)
and (III)
5x - 3y = 8
9x + 3y = 6
14x = 14
\ x = 1
substituting x = 1 in equation (II)
3x + y = 2
\ 3 ´ 1 + y = 2
\ 3 + y = 2
\ y = -1
solution is x = 1, y = -1; it is also
written as (x, y) = (1, -1)
Method (II)
5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Let us write value of y in terms
of x from equation (II) as
y = 2 - 3x . . . (III)
Substituting this value of y in
equation (I).
5x - 3y = 8
\ 5x - 3(2 - 3x) = 8
\ 5x - 6 + 9x = 8
\ 14x - 6 = 8
\ 14x = 8 + 6
\ 14x = 14
\ x = 1
Substituting x = 1 in equation
(III).
y = 2 - 3x
\ y = 2 - 3 ´ 1
\ y = 2 - 3
\ y = -1
x = 1, y = -1 is the solution.
+
3
Ex. (2) Solve : 3x + 2y = 29; 5x - y = 18
Solution : 3x + 2y = 29. . . (I) and 5x - y = 18 . . . (II)
Let’s solve the equations by eliminating ’y’. Fill suitably the boxes below.
Multiplying equation (II) by 2.
\ 5x ´ - y ´ = 18 ´
\ 10x - 2y = . . . (III)
Let’s add equations (I) and (III)
3x + 2y = 29
+
- =
= \ x =
Substituting x = 5 in equation (I)
3x + 2y = 29
\ 3 ´ + 2y = 29
\ + 2y = 29
\ 2y = 29 -
\ 2y = \ y =
(x, y) = ( , ) is the solution.
Ex. (3) Solve : 15x + 17y = 21; 17x + 15y = 11
Solution : 15x + 17y = 21. . . (I)
17x + 15y = 11 . . . (II)
In the two equations above, the coefficients of x and y are interchanged.
While solving such equations we get two simple equations by adding and
subtracting the given equations. After solving these equations, we can easily find
the solution.
Let’s add the two given equations.
15x + 17y = 21
17x + 15y = 11
32x + 32y = 32
+
4
Dividing both sides of the equation by 32.
x + y = 1 . . . (III)
Now, let’s subtract equation (II) from (I)
15x + 17y = 21
17x + 15y = 11
-2x + 2y = 10
dividing the equation by 2.
-x + y = 5 . . . (IV)
Now let’s add equations (III) and (V).
x + y = 1
-x + y = 5
\ 2y = 6 \ y = 3
Place this value in equation (III).
x + y = 1
\ x + 3 = 1
\ x = 1 - 3 \ x = -2
(x, y) = (-2, 3) is the solution.
- - -
Practice Set 1.1
(1) Complete the following activity to solve the simultaneous equations.
5x + 3y = 9 -----(I)
-
+
+
2x - 3y = 12 ----- (II)
Let’s add eqations (I) and (II).
5x + 3y = 9
2x - 3y = 12
x =
x = x =
Place x = 3 in equation (I).
5 ´ + 3y = 9
3y = 9 -
3y =
y =
3
y =
\ Solution is (x, y) = ( , ).
5
2. Solve the following simultaneous equations.
(1) 3a + 5b = 26; a + 5b = 22 (2) x + 7y = 10; 3x - 2y = 7
(3) 2x - 3y = 9; 2x + y = 13 (4) 5m - 3n = 19; m - 6n = -7
(5) 5x + 2y = -3; x + 5y = 4 (6)
1
3
10
3
xy ;
2
1
4
11
4
xy
(7) 99x + 101y = 499; 101x + 99y = 501
(8) 49x - 57y = 172; 57x - 49y = 252
Let’s recall.
Graph of a linear equation in two variables
In the 9
th
standard we learnt that the graph of a linear equation in two
variables is a straight line. The ordered pair which satisfies the equation is a
solution of that equation. The ordered pair represents a point on that line.
Ex. Draw graph of 2x - y = 4.
Solution : To draw a graph of the equation let’s write 4 ordered pairs.
x 0 2 3 -1
y -4 0 2 -6
(x, y) (0, -4) (2, 0) (3, 2) (-1, -6)
To obtain ordered pair by
simple way let’s take x = 0
and then y = 0.
Scale : on both axes
1 cm = 1 unit.
-1 1
1
2
2
3 4 5 6 7 8 9
0
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
(3, 2)
(2, 0)
(0, -4)
(-1, -6)
X
X'
Y
Y'
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FAQs on Textbook: Linear Equations in Two Variables - Mathematics Class 10 (Maharashtra SSC Board)
| 1. What are linear equations in two variables and how can they be represented graphically? |  |
Ans. Linear equations in two variables are mathematical expressions that can be written in the form ax + by + c = 0, where a, b, and c are constants, and x and y are variables. Graphically, these equations represent straight lines on a Cartesian plane. The solutions of these equations correspond to the points where the line intersects the axes or any point on the line itself.
| 2. How do you find the solution of a linear equation in two variables? | |
Ans. To find the solution of a linear equation in two variables, you can use methods such as substitution, elimination, or graphical representation. In substitution, you solve one variable in terms of the other and substitute it back into the original equation. In elimination, you add or subtract equations to eliminate one variable. Graphically, you can plot the equation and identify the points that satisfy it.
| 3. What is the significance of the slope and intercept in the context of linear equations? | |
Ans. The slope of a linear equation indicates the steepness or incline of the line, reflecting how much y changes for a unit change in x. The y-intercept is the point where the line crosses the y-axis, representing the value of y when x is zero. Together, the slope and intercept provide critical information about the line's characteristics and help in sketching the graph of the equation.
| 4. Can a linear equation in two variables have more than one solution? | |
Ans. Yes, a linear equation in two variables can have infinitely many solutions. This occurs because the equation represents a line on the Cartesian plane, and every point (x, y) on that line is a solution. However, there can also be cases where the line does not intersect with another line representing a different equation, leading to no solutions, or they intersect at a single point, leading to one solution.
| 5. How can we use linear equations in real-life situations? | |
Ans. Linear equations in two variables can be applied in various real-life scenarios, such as calculating costs, predicting trends, and solving problems involving relationships between two quantities. For example, they can be used in business to determine profit and loss scenarios, in science to analyze relationships between variables, or in everyday calculations like budgeting or planning trips based on distance and time.
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