Substitution Method

What is a Substitution Method?

  • Substitution method can be defined as a way to solve a linear system algebraically. 

  • The substitution method works by substituting one y-value with the other. The method of substitution involves three steps:

Step 1) First you need to solve one equation for one of the variables.

Step 2) Now you need to substitute (plug-in) this expression into the other equation and solve it.

Step 3) In the last step you need to re-substitute the value into the original equation and you will be able to find the corresponding variable.

Now at first glance, this method might seem complicated, but some helpful tricks will help you to keep things straight. You need to keep in mind that our goal when solving any system is to find the point of intersection. 

What is the Algebraic Method?

An Algebraic method can be defined as a collection of several methods, which are generally used to solve a pair of the linear equations that includes two variables supposedly (a ,b) or (x,y). Generally, the algebraic method can be further subdivided into three different categories:

  1. The Substitution method

  2. The Elimination method

  3. The Cross-multiplication method

What is the Graphical Method?

The graphical method can also be known as the geometric Method used to solve the system of the linear equation. In this graphical method, the equations are designed based on the constraints and objective function .To solve the system of linear equations, this method involves different steps to obtain the solutions.

In this article, we are going to focus mainly on solving the linear equations using the first algebraic method which is known as the “Substitution Method” in detail.
Let’s go through the example,

y=2x+4

3x+y=9

Now we can substitute y in the second equation with the first equation since we can write the second equation in terms of y(y=y)

3x+y=9

y=2x+4

3x + (2x+4) =9

5x +4 =9

5x= 9-4

5x=5

x=1

This value of x that is equal to 1 can then be used to find the value of y by substituting 1 with x example in the first equation,

y=2x+4y

y=2x+4

y=2*1+4y

y=2*1+4

y=6

Therefore, the value of y is equal to 6.

The solution of the linear system is equal to (1, 6).

We can use the substitution method even if both equations of the linear system are in standard form. We can just begin by solving one of the equations for one of its variables.

Questions to be Solved :

Question 1) Solve for the values of ‘x’ and ‘y’:

    x + y = 5.

   3x + y = 11.

Solution) Let’s write down the information given,

        x + y = 5 …………… Equation (i)

       3x + y = 11 …………Equation (ii)

Since we are given two different equations in terms of two different linear equations, now let us try to solve them using the method of substitution:

From the first equation we find that we can write y = 5 - x.

Substituting value of y in the Equation (ii), we get;

 3x + (5 – x) = 11.

2x = 11 - 5

 2x = 6

 x = 6/2

Therefore the value of x = 3.

Now substituting the value x = 3 in the other equation that is y = 5 – x, we get;

y = 5- x 

y = 5 - 3

Therefore, we get the value of y = 2.

Hence, the value of x = 3 and the value of y = 2

Question 2) Solve for the values of ‘x’ and ‘y’:

       2x + 6y = 10

       1x - 2y = 15

Solution) Let’s write down the information given,

       2x + 6y = 10 …………… Equation (i)

      1 x - 2y = 15 …………… Equation (ii)

From the given two equations, let us consider first equation and find out value of one of the variables, say ‘x’ from the equation:

     2x = 10 - 6y

⟹ x = 5 - 3y.

Substituting the value of x = 5 - 3y in Equation (ii), we get;

(5 - 3y) - 2y = 15

⟹ -5y = 15 - 5

⟹ -5y = 10

⟹ y = -2.

Now substituting the value of  y = -2 in the equation x = 5 - 3y, we get;

x = 5 - 3(-2)

⟹ x = 5+6

⟹ x = 11

Hence,  the value of x =11 and y = 2

FAQ (Frequently Asked Questions)

Q1. What are the Steps for the Substitution Method?

Ans. For instance, the system of two equations with two unknown values, we can find the solution using the below steps. Here is the list of steps that you need to know to solve the linear equation. They are as follows:

Step 1) First you need to simplify the given equation by expanding the parenthesis.

Step 2) Now solve one of the equations for either of the variables x or y.

Step 3) Substitute the solution we have got in step 1 in the other equation.

Step 4) Now you need to solve the new equation you have obtained using elementary arithmetic operations.

Step 5) Finally,  you need to solve the equation to find the value of the second variable that is left.

Q2. What is the Substitution Method?

Ans. The method of solving "by substitution" works by solving one of the given equations (you choose which one) for one of the variables (you need to choose which one), and then plugging this value back into the other equation that has been given to you, "substituting" for the chosen variable and solving for the other value. Then you back-solve for the first variable using the first equation.

Q3. Why Does the Substitution Method Work?

Ans. The method of substitution works because you have equality in the objects you're substituting for any given question. If A=B, then you would be able to use B whenever we could use A. So when you have an equation you're free to do operations to both sides of the equation.

Q4. Why is the Substitution Method Better?

Ans. The substitution method is better because we believe that makes solving the equations much easier. Also, depending on the equation, this method involves less work and calculation. This method is the most useful system of two equations to solve two unknowns.