## Linear Equations in Two Variables

Practice Set 1.1

**2. Solve the following simultaneous equations.**

**(1) 3a + 5b = 26; a + 5b = 22**

Ans:-

Equation 1: 3a + 5b = 26

Equation 2: a + 5b = 22

We can solve these equations using the method of substitution or elimination. Let's solve them using the method of substitution:

Step 1: Solve Equation 2 for 'a':

a = 22 - 5b

Step 2: Substitute the value of 'a' in Equation 1:

3(22 - 5b) + 5b = 26

Step 3: Simplify and solve for 'b':

66 - 15b + 5b = 26

-10b = 26 - 66

-10b = -40

b = -40 / -10

b = 4

Step 4: Substitute the value of 'b' back into Equation 2 to find 'a':

a + 5(4) = 22

a + 20 = 22

a = 22 - 20

a = 2

The solution to the simultaneous equations is:

a = 2

b = 4

**(2) x + 7y = 10; 3x - 2y = 7**

Ans:-

Let's solve equation (1) for x:

Equation (1): x + 7y = 10

Step 1: Solve for x

x = 10 - 7y

Now we'll substitute this value of x into equation (2):

Equation (2): 3x - 2y = 7

Step 2: Substitute x = 10 - 7y into equation (2)

3(10 - 7y) - 2y = 7

Simplifying the equation:

30 - 21y - 2y = 7

30 - 23y = 7

-23y = 7 - 30

-23y = -23

y = -23 / -23

y = 1

Now that we have the value of y, we can substitute it back into equation (1) to find the value of x:

x + 7(1) = 10

x + 7 = 10

x = 10 - 7

x = 3

Therefore, the solution to the simultaneous equations is x = 3 and y = 1.

**(3) 2x - 3y = 9; 2x + y = 13**

Ans:-

Let's solve for x in the second equation and substitute it into the first equation.

Equation 1: 2x - 3y = 9 ---- (1)

Equation 2: 2x + y = 13 ---- (2)

Solving Equation 2 for x:

2x = 13 - y

x = (13 - y) / 2

Substituting the value of x in Equation 1:

2((13 - y) / 2) - 3y = 9

13 - y - 3y = 9

13 - 4y = 9

-4y = 9 - 13

-4y = -4

y = -4 / -4

y = 1

Substituting the value of y in Equation 2 to find x:

2x + 1 = 13

2x = 13 - 1

2x = 12

x = 12 / 2

x = 6

Therefore, the solution to the simultaneous equations is x = 6 and y = 1.

**(4) 5m - 3n = 19; m - 6n = -7**

Ans:-

Equation 1: 5m - 3n = 19 ----(1)

Equation 2: m - 6n = -7 ----(2)

Step 1: Solve Equation 2 for m:

From Equation 2, we have: m = 6n - 7

Step 2: Substitute the value of m into Equation 1:

5(6n - 7) - 3n = 19

30n - 35 - 3n = 19

27n - 35 = 19

27n = 54

n = 54 / 27

n = 2

Step 3: Substitute the value of n into Equation 2 to find m:

m - 6(2) = -7

m - 12 = -7

m = -7 + 12

m = 5

So, the solution to the given system of equations is:

m = 5, n = 2

**(5) 5x + 2y = -3; x + 5y = 4**

Ans:-

Let's solve the first equation for x:

Equation 1: 5x + 2y = -3

Solving for x:

5x = -2y - 3

x = (-2y - 3) / 5

Now, we substitute this expression for x into the second equation:

Equation 2: x + 5y = 4

Substituting x = (-2y - 3) / 5:

((-2y - 3) / 5) + 5y = 4

Now, we can solve this equation for y:

((-2y - 3) / 5) + 5y = 4

Multiply through by 5 to eliminate the fraction:

-2y - 3 + 25y = 20

23y - 3 = 20

23y = 23

y = 1

Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use Equation 1:

5x + 2y = -3

5x + 2(1) = -3

5x + 2 = -3

5x = -5

x = -1

Therefore, the solution to the given system of equations is x = -1 and y = 1.

**(6) 1/3 x + y= 10/3 ; 2x+ 1/4y=11/4**

Ans:-

Equation 1: 1/3x + y = 10/3

Equation 2: 2x + 1/4y = 11/4

Step 1: Solve Equation 1 for x:

1/3x = 10/3 - y

x = (10/3 - y) * 3/1

x = 10 - 3y

Step 2: Substitute x into Equation 2:

2(10 - 3y) + 1/4y = 11/4

20 - 6y + 1/4y = 11/4

Step 3: Simplify and solve for y:

Multiplying the equation by 4 to eliminate the fraction:

80 - 24y + y = 11

-23y = 11 - 80

-23y = -69

y = -69 / -23

y = 3

Step 4: Substitute y back into Equation 1 to solve for x:

1/3x + 3 = 10/3

1/3x = 10/3 - 3

1/3x = 10/3 - 9/3

1/3x = 1/3

x = 1/3 * 3/1

x = 1

Therefore, the solution to the simultaneous equations is x = 1 and y = 3.

**(7) 99x + 101y = 499; 101x + 99y = 501**

Ans:- Given equations:

Equation (1): 99x + 101y = 499

Equation (2): 101x + 99y = 501

Adding equations (1) and (2):

(99x + 101y) + (101x + 99y) = 499 + 501

200x + 200y = 1000

Dividing the equation by 200:

x + y = 5

Subtracting equation (1) from equation (2):

(101x + 99y) - (99x + 101y) = 501 - 499

2x - 2y = 2

Dividing the equation by 2:

x - y = 1

Now we have a system of equations:

x + y = 5

x - y = 1

Adding these equations:

2x = 6

Dividing by 2:

x = 3

Substituting the value of x into one of the equations (e.g., x - y = 1):

3 - y = 1

Solving for y:

-y = 1 - 3

-y = -2

y = 2

Therefore, the solution to the simultaneous equations is:

x = 3

y = 2

**(8) 49x - 57y = 172; 57x - 49y = 252**

Ans:-

Equation (III): x - y = 4

Equation (IV): x + y = 10

To eliminate y, let's add Equation (III) and Equation (IV):

(x - y) + (x + y) = 4 + 10

2x = 14

x = 14/2

x = 7

Now substitute the value of x back into Equation (IV):

7 + y = 10

y = 10 - 7

y = 3

Therefore, the solution to the given system of equations is x = 7 and y = 3.

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