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Solve the Linear Equation using Substitution Method

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The equation in which the highest power of the variable is always 1 is called a linear equation (or) the first-order equation. A linear equation’s graph will always be a straight line. When the equation has only one variable and the highest degree is 1, then it is called a linear equation in one variable. The general form of a linear equation in one variable is ax + b = 0, where a is a coefficient of x and b is constant. The standard form of a linear equation in two variables is of the form ax + by + c = 0, where a and b are coefficients of x and y, respectively, and c is a constant. Some of the examples of linear equations are 3x+4= 0, 2y = 8, m + n = 5, 4a – 3b + c = 7, x/2 = 8, etc. There are mainly two methods for solving simultaneous linear equations: the graphical method and the algebraic method. The algebraic method is further classified into three types, namely: 

  1. Substitution method
  2. Elimination method
  3. Cross-multiplication method

Substitution Method

Solving a linear equation means finding the solution to the given linear equation. There are mainly two methods for solving simultaneous linear equations: the graphical method and the algebraic method. The substitution method is one of the algebraic methods to solve a system of linear equations with two variables. As the name implies, in the substitution method, the value of a variable from one equation is substituted into the second equation. Thus, a pair of linear equations gets transformed into one linear equation with just one variable, which can then be solved with ease. For example, we have to find the value of the x-variable in terms of the y-variable from the first equation and then substitute the value of the x-variable in the second equation. Thus, we can solve and find the value of the y-variable. Finally, we can substitute the value of y in any of the given equations to find x. We can also interchange this process, where we first solve for x and then solve for y.

Steps to solve a System of Equations by Substitution Method

The following are the steps that are applied while solving a system of equations by using the Substitution Method.

  • Step 1: If necessary, expand the parentheses to simplify the given equation.
  • Step 2: Solve one of the given equations for any of the variables. Depending upon the ease of calculation, you can use any variable.
  • Step 3: Now, substitute the solution obtained from step 2 into the other equation.
  • Step 4: Now, simplify the new equation obtained by using the fundamental arithmetic operations and solve the equation for one variable.
  • Step 5: Finally, to find the value of the second variable, substitute the value of the variable obtained from step 4 into any of the given equations.

Now,
Let us go through an example of solving a system of equations by using the substitution method, 3(x+4)−6y = 0 and 5x+3y+7 = 0.

Solution:

Step 1:

By simplifying the first equation further, we get 3x − 6y + 12 = 0.

Now, the two equations are:

3x − 6y + 12 = 0  ———— (1)

5x + 3y + 7 = 0 ———— (2)

Step 2: 

By solving equation (1),   x = (−12 + 6y)/3 = −4 + 2y 

Step 3: 

Substitute the value of x obtained equation (2). i.e., we are substituting x = −4 + 2y  in the equation 5x + 3y + 7 = 0.

5(−4 + 2y) + 3y + 7 = 0

Step 4: 

Now, simplify the new equation obtained in the above step.

⇒ 5(−4 + 2y) + 3y + 7 = 0

⇒ −20 + 10y + 3y + 7 = 0

⇒ 13y − 13 = 0

⇒ 13y = 13

⇒ y = 13/13 ⇒ y = 1

Step 5: 

Now, substitute the value of y obtained in any of the given equations. Let us substitute the value of y in equation (1).

⇒ 3x − 6y + 12 = 0

⇒ 3x −6(1) + 12 = 0

⇒ 3x −6 + 12 = 0

⇒ 3x + 6 = 0

⇒ 3(x + 2) = 0

⇒ x + 2 = 0 ⇒ x = −2

Thus, by solving the given system of equations using the substitution method, we get x = −2 and y= 1.

Difference between Substitution Method and Elimination Method

The substitution method and the elimination method are algebraic methods for solving simultaneous linear equations. Now, let’s go through the differences between the two methods.

 Substitution Method 

Elimination Method

In the substitution method, the value of a variable from one equation is substituted into the second equation. 

In the elimination method, we make the coefficient of either x-variable or y-variable of both the equations the same by multiplying or dividing either one or both of the equations by a number. By adding or subtracting from both equations, the variable whose coefficient is the same is eliminated. Thus, the value of one variable is found, which can be substituted in any one of the equations to determine the other variable too. 

The substitution method can be applied easily to equations involving smaller values or when the given equations are in the form of x = ay + b and y = mx + n. 

The elimination method can be applied easily to equations involving large numbers or fractions compared to the substitution method. When the coefficient of any one of the terms is the same, it is preferable to use the elimination method. For example, we can use the elimination method when ax + by + c = 0 and px + by + r = 0.

Solved Examples based on Substitution Method

Example 1: Solve: 4x−3y = 5 and 3x + y = 7, using the substitution method.

Solution: 

The given two equations are:

4x−3y = 5    ————(1)

3x + y = 7    ————(2)

Now, the solution to the given two equations can be found by the following steps:

From equation (2) we can find the value of y in terms of x, i.e.,

y = 7 − 3x

Now, substitute the value of y in equation (1). 

⇒  4x − 3(7−3x) = 5

⇒ 4x − 21+ 9x = 5

⇒ 13x = 21 + 5

⇒ 13x = 26

⇒ x = 26/13 = 2

Substitute the value of x in equation 2, 

⇒ 3(2) + y = 7

⇒ y = 7 − 6 = 1

Hence, the values of x and y are 2 and 1, respectively.

Example 2: Solve: 2m + 5n = 1 and 3m − 2n = 11 by using substitution method.

Solution: 

The given two equations are:

2m + 5n = 1  ————(1)

3m − 2n = 11   ————(2)

Now, the solution to the given two equations can be found by the following steps:

From equation (2) we can find the value of m in terms of n, i.e.,

m = (11 + 2n)/3  ————(3)

Now, substitute the value of m in equation (1). 

⇒ 2[(22 + 2n)/3] + 5n = 1

⇒ (22 + 4n)/3 + 5n = 1

⇒ [(22 + 4n) + 15n]/3 = 1

⇒ 22 + 19n = 3

⇒ 19n = 3 − 22 = −19

⇒ n = −19/19 = −1

Substitute the value of n in equation 3, 

⇒ m = (11 + 2(−1))/3

⇒ m = (11−2)/3

⇒ m = 9/3 = 3

Hence, the values of m and n are 3 and −1, respectively.

Example 3: Solve 6a − 4b = 15 and 2a + 3b = −8 by using substitution method.

Solution:

The given two equations are:

6a − 4b = 15  ————(1)

2a + 3b = −8   ————(2)

Now, the solution to the given two equations can be found by the following steps:

From equation (2) we can find the value of “a” in terms of b, i.e.,

a = (−8 − 3b)/2  ————(3)

Now, substitute the value of “a” in equation (1). 

⇒ 6[(−8 − 3b)/2) − 4b = 15

⇒ (−48 − 18b)/2 − 4b = 15

⇒ −48 − 18b − 8b = 15 × 2

⇒ −48 − 26b = 30

⇒ −26b = 30 + 48 = 78

⇒ b = −78/26 = −3

Now substitute the value of “b” in equation (3)

⇒ a = (−8 −3(−3))/2

⇒ a = (−8 + 9)/2 = 1/2

⇒ a = 0.5

Hence, the values of a and b are 0.5 and −3, respectively.

Example 4: If the sum of the two numbers is 38 and the difference between them is 12. Find the numbers using the substitution method.

Solution:

Let the two numbers be x and y.

From the given data, we can write

x + y = 38   ————(1)

x − y = 12   ————(2)

Now, the solution to the given two equations can be found by the following steps:

From equation (2) we can find the value of x in terms of y, i.e.,

x = 12 + y  ————(3)

Now, substitute the value of x in equation (1). 

⇒ 12 + y + y = 38

⇒ 12 + 2y = 38

⇒ 2y = 38 − 12 = 26

⇒ y = 26/2 = 13

 Now substitute the value of y in equation (3)

⇒ x = 12 + 13 = 25

Hence, the two given numbers are 25 and 13.

Example 5: Solve: m + n = 5 and 4m − 3n = 6 by using substitution method.

Solution:

The given two equations are:

m + n = 5  ————(1)

4m − 3n = 6   ————(2)

Now, the solution to the given two equations can be found by the following steps:

From equation (1) we can find the value of m in terms of n, i.e.,

m = 5 − n ————(3)

Now, substitute the value of m in equation (2).

⇒ 4(5−n) − 3n = 6

⇒ 20 − 4n − 3n = 6

⇒ 20 − 7n =6

⇒ 20 − 6 = 7n

⇒ 7n = 14

⇒ n = 14/7 ⇒ n = 2

Substitute the value of n in equation 1,

⇒ m + 2 = 5

⇒ m = 5 − 2 ⇒ m = 3

Hence, the values of m and n are 3 and 2, respectively.

FAQs on Substitution Method

Question 1: What is meant by a linear equation?

Answer: 

The equation in which the highest power of the variable is always 1 is called a linear equation (or) the first-order equation. A linear equation’s graph will always be a straight line. Some of the examples of linear equations are 3x+4= 0, 2y = 8, m + n = 5, 4a – 3b + c = 7, x/2 = 8, etc. 

Question 2: What are the different methods to solve the system of equations linear equations in two variables?

Answer:

There are mainly two methods for solving simultaneous linear equations: the graphical method and the algebraic method. The algebraic method is further classified into three types namely:

  • Substitution method
  • Elimination method
  • Cross-multiplication method

Question 3: What is meant by the substitution method in algebra?

Answer: 

The substitution method is one of the algebraic methods to solve a system of linear equations with two variables. As the name implies, in the substitution method, the value of a variable from one equation is substituted into the second equation. Thus, a pair of linear equations gets transformed into one linear equation with just one variable, which can then be solved with ease.

Question 4: What is the first step in the substitution method?

Answer:

The first step in the substitution method is to solve one of the given equations for any of the variables. Depending upon the ease of calculation, you can use any variable. For example, if there are two equations, x+2y = 7 and x-y = 1, by applying the first step of the substitution method, we get x = 1+y from the second equation.

Question 5: What are the steps used in the Substitution Method?

Answer: 

The following are the steps that are applied while solving a system of equations by using the Substitution Method.

Step 1: If necessary, expand the parentheses to simplify the given equation.
Step 2: Solve one of the given equations for any of the variables. Depending upon the ease of calculation, you can use any variable.
Step 3: Now, substitute the solution obtained from step 2 into the other equation.
Step 4: Now, simplify the new equation obtained by using the fundamental arithmetic operations and solve the equation for one variable.
Step 5: Finally, to find the value of the second variable, substitute the value of the variable obtained from step 4 into any of the given equations.


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Last Updated : 06 Jan, 2023
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