SOLVING LINEAR EQUATIONS USING ELIMINATION METHOD

Step 1 :

In the given two linear equations, make sure that one of the variables having the same coefficient. If not, using least common multiple, make the coefficient of one of the variables is same in both the equations.

Step 2 :

When one of the variables has the same coefficient in both equations, make sure that they have different signs. So that, when the equations are added, the variable can be eliminated.

Step 3 :

Once a variable is eliminated, solve for the other variable.

Step 4 :

Using the value of the variable in step 3, find the value of the eliminated variable by substituting the value in one of the equations.

Example 1 :

2x + 3y = 5

x - 3y = -2

Solution :

2x + 3y = 5 ----(1)

x - 3y = -2 ----(2)

(1) + (2) :

3x = 3

Divide both sides by 3.

x = 1

Substitute x = 1 in (1).

2(1) + 3y = 5

2 + 3y = 5

Subtract 2 from both sides.

3y = 3

Divide both sides by 3.

y = 1.

Therefore,

x = 1 and y = 1

Example 2 :

3x - 2y = 1

5x - 3y = 3

Solution :

3x - 2y = 1 ----(1)

5x - 3y = 3 ----(2)

In the above equation, none of the variables have the same coefficient in both the equations.

Let's eliminate the variable y. Consider the coefficients of y in the above two equations. The coefficients of y are 2 and 3 (without considering the sign).

Least common multiple of (2, 3) = 6.

Make the coefficient of y as 6 in both the equations multiplying by suitable factors.

And also, when you multiply, make sure that the y-terms have different signs in the two equations. So that, when you add the equations, y-terms can be eliminated.

3x(1):

9x - 6y = 3 ----(3)

-2x(2):

-10x + 6y = -6 ----(4)

(3) + (4) :

-x = -3

Multiply both sides by -1.

x = 3.

Substitute x = 3 in (1).

3(3) - 2y = 1

9 - 2y = 1

Subtract 9 from both sides.

-2y = -8

Divide both sides by -2.

y = 4.

Therefore,

x = 3 and y = 4

Example 3 :

1/2x + 1/3y = 2

1/3x + 1/2y = 13/6

Solution :

Let a = 1/x and b = 1/y.

1/2x + 1/3y = 2 :

a/2 + b/3 = 2

Multiply both sides by 6.

6(a/2 + b/3) = 6(2)

6(a/2) + 6(b/2) = 12

3a + 2b = 12 ----(1)

1/3x + 1/2y = 13/6 :

a/3 + b/2 = 13/6

Multiply both sides by 6.

6(a/3 + b/2) = 6(13/6)

6(a/3) + 6(b/2) = 13

2a + 3b = 13 ----(2)

3x(1) :

9a + 6b = 36 ----(3)

-2x(2) :

-4a - 6b = -26 ----(4)

(3) + (4) :

5a = 10

Divide both sides by 5.

a = 2

Substitute a = 2 in (1).

3(2) + 2b = 12

6 + 2b = 12

Subtract 6 from both sides.

2b = 6

Divide both sides by 2.

b = 3

Solve for x and y :

1/x = a

1/x = 2

x = 1/2

1/y = b

1/y = 3

y = 1/3

Example 4 :

2/√x + 3/√y = 2

4/√x – 9/√y = -1

Solution :

Let a = 1/√x and b = 1/√y.

2/√x + 3/√y = 2 :

2a + 3b = 2 ----(1)

4/√x – 9/√y = -1 :

4a – 9b = -1 ----(2)

3x(1) :

6a + 9b = 6 ----(3)

(2) + (3) :

10a = 5

Divide both sides by 10.

a = 1/2

2(1/2) + 3b = 2

1 + 3b = 2

Subtract 1 from both sides.

3b = 1

Divide both sides by 3.

b = 1/3

Solve for x and y :

1/√x = 1/2

√x = 2

x = 22

x = 4

 1/√y  = 1/3

√y = 3

y = 32

y = 9

Example 5 :

4/x + 3y = 14

3/x – 4y = 23

Solution :

Let a = 1/x and b = y.

4/x + 3y = 14 :

4a + 3b = 14 ----(1)

3/x – 4y = 23 :

3a – 4b = 23 ----(2)

4x(1) :

16a + 12b = 56 ----(3)

3x(2) :

9a – 12b = 69 ----(4)

(3) + (4) :

25a = 125

Divide both sides by 25.

a = 5

Substitute a = 5 in (1).

4(5) + 3b = 14

20 + 3b = 14

Subtract 20 from both sides.

3b = -6

Divide both sides by 3.

b = -2

1/x = a

1/x = 5

x = 1/5

y = b

y = -2

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