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Substitution Method

One of the methods to solve a system of linear equations in two variables algebraically is the "substitution method". In this method, we find the value of any one of the variables by isolating it on one side and taking every other term on the other side of the equation. Then we substitute that value in the second equation.

The substitution method is preferable when one of the variables in one of the equations has a coefficient of 1. It involves simple steps to find the values of variables of a system of linear equations by substitution method. Let's learn about it in detail in this article.

1.	What is Substitution Method?
2.	Solving Systems of Equations by Substitution Method
3.	Difference Between Elimination and Substitution Method
4.	FAQs on Substitution Method

What is Substitution Method?

The substitution method is a simple way to solve a system of linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of the x-variable in terms of the y-variable from the first equation and then substituting or replacing the value of the x-variable in the second equation. In this way, we can solve and find the value of the y-variable. And at last, we can put the value of y in any of the given equations to find x This process can be interchanged as well where we first solve for x and then solve for y.

In simple words, the substitution method involves substituting the value of any one of the variables from one equation into the other equation. Let us take an example of solving two equations x-2y=8 and x+y=5 using the substitution method.

solving system of linear equations by substitution method gives the result as x equals 6 and y equals minus 1

☛ Note: The other three algebraic methods of solving linear equations. To learn each of these methods, click on the respective links given below.

Solving Systems of Equations by Substitution Method

The steps to apply or use the substitution method to solve a system of equations are given below:

Step 1: Simplify the given equation by expanding the parenthesis if needed.
Step 2: Solve any one of the equations for any one of the variables. You can use any variable based on the ease of calculation.
Step 3: Substitute the obtained value of x or y in the other equation.
Step 4: Now, simplify the new equation obtained using arithmetic operations and solve the equation for one variable.
Step 5: Now, substitute the value of the variable from Step 4 in any of the given equations to solve for the other variable.

Here is an example of solving system of equations by using substitution method: 2x+3(y+5)=0 and x+4y+2=0.

Solution:

Step 1: Simplify the first equation to get 2x + 3y + 15 = 0. Now we have two equations as,

2x + 3y + 15 = 0 _____ (1)

x + 4y + 2 = 0 ______ (2)

Step 2: We are solving equation (2) for x. So, we get x = -4y - 2.

Step 3: Substitute the obtained value of x in the equation (1). i.e., we are substituting x = -4y-2 in the equation 2x + 3y + 15 = 0, we get, 2(-4y-2) + 3y + 15 = 0.

Step 4: Now, simplify the new equation. We get, -8y-4+3y+15=0

-5y + 11 = 0

-5y = -11

y = 11/5

Step 5: Now, substitute the value of y in any of the given equations. Let us substitute the value of y in equation (2).

x + 4y + 2 = 0

x + 4 × (11/5) + 2 = 0

x + 44/5 + 2 = 0

x + 54/5 = 0

x = -54/5

Therefore, after solving the given system of equations by substitution method, we get x = -54/5 and y= 11/5.

Difference Between Elimination and Substitution Method

Both elimination and substitution methods are ways to solve linear equations algebraically. When the substitution method becomes a little difficult to apply in equations involving large numbers or fractions, we can use the elimination method to ease our calculations. Let us understand the difference between these two methods through the table given below:

Substitution Method	Elimination Method
In this, we find the value of any one of the variables and substitute its value in the other equation.	In this method, we multiply or divide either one or both equations by a number to make the coefficient of either x-variable or y-variable the same in both equations. Then, we add or subtract the equations to eliminate the variable whose coefficient is the same. In this way, we find the value of one variable which can be substituted in any one of the equations to find the other variable too.
It is better to use the substitution method when equations are either given in form or can be gotten into the form of x = ay + b and y = mx + n.	It is better to use the elimination method when the coefficients of none of the variables are the same.
In this method, one variable must be isolated in at least one of the equations.	This method helps to solve the system without isolating any of the variables.

Important Notes on Substitution Method:

To start with the substitution method, first, select the equation that has coefficient 1 for at least one of the variables and solve for the same variable (with coefficient 1). This makes the process easier.
Before starting with the substitution method, combine all like terms (if any).
After solving for one variable, we can select any of the given equations or any equation in the whole process to find the other variable.
Determining infinite/no solutions by substitution method:
- If we get any true statement like 3 = 3, 0 = 0, etc while solving using the substitution method, then it means that the system has infinitely many solutions.
- If we get any false statement like 3 = 2, 0 = 1, etc then the system has no solution.

☛ Related Topics:

Substitution Method Examples

Example 1: Solve the system of linear equations by substitution method: 5m−2n=17 and 3m+n=8.

Solution:

The given two equations are:

5m−2n=17 ____ (1)

3m+n=8 _____ (2)

The solution of the given two equations can be found by the following steps:
- From equation 2 we can find the value of n in terms of m, where n = 8 - 3m
- Substitute the value of n in equation 1. We get, 5m - 2(8-3m)=17
5m - 2(8-3m)=17

5m - 16 + 6m =17

11m = 17 + 16

11m=33

m = 3
- Substitute the value of m in equation 2, we get, 3×3+n=8
9+n=8

n=8-9

n=-1

Answer: ∴ The solution is m=3 and n=-1.
Example 2: Jacky has two numbers such that the sum of two numbers is 20 and the difference between them is 10. Find the numbers.

Solution:

Let the two numbers be x and y such that x>y. It is given that,

x+y=20 ___ (1)

and x−y=10 ___ (2).

We will now solve by substitution.

From equation 1, we get x = 20-y. Substitute this value in equation 2 to find the value of y.

x−y=10

20-y-y=10

20-2y=10

20-10=2y

10=2y

y=10/2 = 5

Now, substitute the value of y in equation 1, we get x+5=20, which gives us x=15.

Answer: Therefore, the two numbers are 15 and 5.
Example 3: Solve the given system of linear equations by substitution method:

- 2x - 5 + 3x + y = 0 ___ (1)

3x + y = 11 ___ (2)

Solution:

As we can see that the first equation can be further simplified by combining like terms. After simplifying it, we get x+y-5=0. From this equation, let us find the value of x in terms of y, which is x = 5-y. Now substitute this value in equation 2, we get 3(5-y)+y=11.

15-3y+y=11

15-2y=11

15-11=2y

4=2y

y=4/2=2

Now, let us substitute the value of y in equation 1. We get x+2-5=0, which can be simplified to x = 3.

Answer: ∴ Solving systems by substitution, x=3 and y=2.

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FAQs on Substitution Method

What is the Substitution Method in Algebra?

In algebra, the substitution method is one of the ways to solve linear equations in two variables. In this method, we substitute the value of a variable found by one equation in the second equation. It is very easy to use when we have smaller numbers, but in the case of large numbers or fractional coefficients, it becomes tedious to apply the substitution method.

What are the Steps for the Substitution Method?

The three simple steps for the substitution method are given below:

Find the value of any one variable from any of the equations in terms of the other variable.
Substitute it in the other equation and solve.
Substitute the value of the second variable again in any of the equations.

When would you Use the Substitution Method?

The substitution method can be applied to any pair of linear equations with two variables. It is advisable to use the substitution method when we have smaller coefficients in terms or when the equations are given in form x = ay+c and/or y=bx+p.

What do we Substitute in the Substitution Method?

In the substitution method, we substitute the value of one variable found by simplifying an equation in the other equation. For example, if there are two variables in the equations, say m and n, then we can first find the value of m in terms of n from any one of the equations, and then we substitute that value in the second equation to get an answer of n. Then, we again substitute the value of n in any of the given equations to find m.

What do the Substitution Method and the Elimination Method have in Common?

Both methods involve the process of substitution. In both methods, we find the value of one variable first and then substitute it in any of the given equations.

What is the First Step in the Substitution Method?

The first step in the substitution method is to find the value of any one of the variables from one equation in terms of the other variable. For example, if there are two equations x+y=7 and x-y=8, then from the first equation we can find that x=7-y. The further steps involve substituting this in the other equation and then solving.

What is the Process of Solving Systems by Substitution?

With a system of equations with variables x and y, we first find the value of x in terms of y from any one of the equations given. Then, we substitute that value in the other equation to find the value of y. At last, we again substitute the value of y in any given equation to find x.

Is the Substitution Method Only for Linear Equations?

No, substitution method can be applied for any type of equations. For example, the equations y = x² and y = 3x + 4 can be solved by using the substitution method.

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