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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