# Substitution Method

Observe the steps how to solve the system of linear equations by using the substitution method.

(i) Find the value of one variable in terms of the other from one of the given equations.

(ii) Substitute the value of this variable in the other equation.

(iii) Solve the equation and get the value of one of the variables.

(iv) Substitute the value of this variable in any of the equation to get the value of other variable.

Follow the instructions along with the method of solution of the two simultaneous equations given below to find the value of x and y.

7x – 3y = 31 --------- (i)

9x – 5y = 41 --------- (ii)

Step I:

From equation (i) 7x – 3y = 31, express y in terms of x

From equation (i) 7x – 3y = 31, we get;

– 3y = 31 – 7x

or, 3y = 7x – 31

or, 3y/3 = (7x – 31)/3

Therefore, y = (7x – 31)/3 --------- (iii)

Step II:

Substitute the value of y obtained from equation (iii) (7x – 31)/3 in equation (ii) 9x – 5y = 41

Putting the value of y obtained from equation (iii) in equation (ii) we get;

9x – 5 × (7x – 31)/3 = 41 --------- (iv)

Step III:

Now, solve equation (iv) 9x – 5 × (7x – 31)/3 = 41

Simplifying equation (iv) 9x – 5 × (7x – 31)/3 = 41 we get;

(27x – 35x + 155)/3 = 41

or, 27x – 35x + 155 = 41 × 3

or, 27x – 35x + 155 = 123

or, –8x + 155 = 123

or, –8x + 155 – 155 = 123 – 155

or, –8x = –32

or, 8x/8 = 32/8

Therefore, x = 4

Step IV:

Putting the value of x in equation (iii)

y = (7x – 31)/3, find the value of y

Putting x = 4 in equation (iii), we get;

y = (7 × 4 – 31)/3

or, y = (28 – 31)/3

or, y = –3/3

Therefore, y = –1

Step V:

Write down the required solution of the two simultaneous linear equations by using the substitution method

Therefore, x= 4 and y = –1

In this case, the general method obtained for solving simultaneous equations as follows:

1. To express y in terms of x from any one of the equations.

2. To substitute this value of y in the other equation.

3. One value of x will be obtained, by solving the equation in x thus obtained.

4. Substituting this value of x in any of the equations, we will get the corresponding value of y.

5. Solution of the two given simultaneous equations will be given by this pair of values of x and y.

6. Similarly expressing x in terms of y from an equation and substituting in the other, we can find the value of y. Putting this value of in any one of the equations, we can find the value of x and thus we can solve the two linear simultaneous equations.

As in this method of solution, we express one unknown quantity in terms of the other and substitute in an equation; o we call this method as ‘Method of Substitution’.

Keep these instructions in your mind and notice how the following simultaneous equations can be solved.

Worked-out examples on two variables linear equations by using the substitution method:

2/x + 3/y = 2 --------- (i)

5/x + 10/y = 5⁵/₆ --------- (ii)

From equation (i), we get:

3/y = 2 – 2/x

or, 3/y = (2x – 2)/x

or, y/3 = x/(2x – 2)

or, y = 3x/(2x – 2) --------- (iii)

Substituting 3x/(2x – 2) in place of y in equation (ii),

or, 5/x + 10 ÷ 3x/(2x – 2) = 35/6

or, 5/x + 10(2x – 2)/3x = 35/6

or, 1/x + 2(2x – 2)/3x = 7/6

or, (3 + 4x – 4)/3x = 7/6

or, (4x – 1)/3x = 7/6

or, (4x – 1)/x = 7/2

or, 8x – 2 = 7x

or, 8x – 2 + 2 = 7x + 2 or, 8x – 7x = 7x – 7x + 2

or, x = 2

Putting the value of x = 2 in equation (iii),

or, y = 3 ∙ 2/2 ∙ 2 – 2

or, y = 6/4 – 2

or, y = 6/2

or, y = 3

Therefore, the required solution is x = 2 and y = 3.

Simultaneous Linear Equations

Simultaneous Linear Equations

Comparison Method

Elimination Method

Substitution Method

Cross-Multiplication Method

Solvability of Linear Simultaneous Equations

Pairs of Equations

Word Problems on Simultaneous Linear Equations

Word Problems on Simultaneous Linear Equations

Practice Test on Word Problems Involving Simultaneous Linear Equations

Simultaneous Linear Equations - Worksheets

Worksheet on Simultaneous Linear Equations

Worksheet on Problems on Simultaneous Linear Equations

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Method of H.C.F. |Highest Common Factor|Factorization &Division Method

Apr 13, 24 05:12 PM

We will discuss here about the method of h.c.f. (highest common factor). The highest common factor or HCF of two or more numbers is the greatest number which divides exactly the given numbers. Let us…

2. ### Factors | Understand the Factors of the Product | Concept of Factors

Apr 13, 24 03:29 PM

Factors of a number are discussed here so that students can understand the factors of the product. What are factors? (i) If a dividend, when divided by a divisor, is divided completely

3. ### Methods of Prime Factorization | Division Method | Factor Tree Method

Apr 13, 24 01:27 PM

In prime factorization, we factorise the numbers into prime numbers, called prime factors. There are two methods of prime factorization: 1. Division Method 2. Factor Tree Method

4. ### Divisibility Rules | Divisibility Test|Divisibility Rules From 2 to 18

Apr 13, 24 12:41 PM

To find out factors of larger numbers quickly, we perform divisibility test. There are certain rules to check divisibility of numbers. Divisibility tests of a given number by any of the number 2, 3, 4…