Substitution Method

James went for grocery shopping at a nearby store. He bought 2 boxes of chocolate and 1 packet of bread for $70. The following week, he bought 1 box of chocolate and 2 packets of bread for $75. He wants to know the price per item for the chocolate box and a packet of bread.

system of equations substitution method

Can we help James solve his query?

To solve his problem, James needs to apply a method of solving linear equations in two variables. This page is exactly what he is looking for.

In this chapter, we will learn about solving linear equations in two variables using the substitution method. 

Check out the interactive simulation on the method to solve the system of equations- substitution method using substitution method calculator.

Do not forget to try your hand at a few interactive questions at the end of the page. 

Lesson Plan

Solving Linear Equations By Substitution Method 

How to Do the Substitution Method

The solution of the linear equation using the substitution method can be found out by following the steps given below:

  • Let us assume two linear equations in the general form:

 linear equations in two variables

  • Simplify the given equation by expanding the parenthesis if needed.
  • Solve any one of the equations for any one of the variables. You can use any variable based on the ease of calculation.

solving linear equations-examples

Here, we have used equation 2 to calculate the value of variable \( x\). 

solving linear equations in two variables-example

  • Substitute the obtained value of \( x \) in the other equation.

systems of equations substitution method-example

  • Now, simplify the new equation obtained using elementary arithmetic operations.

substitution method algebra-final image

Finally, simplify the equation in terms of the second variable.

Let us take an example.

Examples

If we are given two equations:

\( \begin{align*} 4m - 3n &= 32 \rightarrow 1 \\  m + n &=1 \rightarrow 2 \end{align*} \)

Find the value of m and n.

Solution:

  • Using the second equation, find the value of m in terms of n, i.e.,

\[\begin{align*} m+n&=1\\  n&=1-m\end{align*}\]

  • Substuting the value of n in the first equation to find the value of m, we get,

\( 4m-3(1-m)=32\)

  • Simplify the equation in terms of m and find the value of m.

\[\begin{align*}4m-3+3m&=32\\ 7m &=32+3\\ 7m&=35\\ 7m &=35\end{align*}\]

  • Substituting the value of m in Eq. (1):

\[\begin{align*} n = (1 - m) &= (1 - 5)\\  n &= -4\end{align*}\]

\(\therefore\) Solution of given pair of equation:

  m = 5, n = - 4


The Difference Between the Substitution Method And Elimination Methods

The difference between the substitution method and elimination methods are listed in the table below:

Substitution Method Elimination Method
The substitution method consists of solving a variable in the function of another variable and then substituting the value of the first variable into another equation, due to which equation becomes the function of a single variable. the value of the variables can be found after simplification.  The elimination method consists of eliminating the same variable from two equations and then we find the value of another variable that was not eliminated after taking the difference of two equations. Then the obtained value of one variable is placed in either of the two equations. further, the value of another variable can be found after simplification.
If the coefficient of any variable is 1, in any one of the equation, then it is easier to use the substitution method.  The elimination method is used when the coefficients of variables are not equal to 1.

Substitution Method Calculator

Check out the following substitution method calculator to solve any pair of linear equations in two variables.


Solved Examples

Example 1

 

 

Sean was given two equations \(5m-2n=17\) and \(3m+n=8\), can you how help him in finding the solution of these equations using the substitution method?

Solution

We have,

\( \begin{align*} 5m-2n &= 17 \rightarrow 1 \\ 3m+n &=8 \rightarrow 2 \end{align*} \)

The solution of the given two equations can be found by following steps: 

  •  Using the second equation, find the value of m in terms of n, i.e.,

\[\begin{align*} 5m-2n&=17\\ 5m&=17+2n\\  m&= \dfrac{17+2n}{5}\end{align*}\]

  • Substuting the value of m in the first equation to find the value of n.

\( \dfrac{3(17+2n)}{5} +n=8 \)

  • Simplify the equation in terms of n and find the value of n.

\[\begin{align*}3(17+2n)+5n&=8\times5\\  51+6n+5y&=40\\ 11n&=40-51\\ 11n&=-11\\ n&=-1\end{align*}\]

  • Substituting the value of n in Eq. (1), we have,

\[\begin{align*}m&= \dfrac{17+2n}{5}\\ m&= \dfrac{17+2(-1)}{5}\\ m&= 3\end{align*}\]

\(\therefore\) Solution of given pair of equation:

m = 3

    n = -1 

Example 2

 

 

Jacky has two numbers, the sum of two numbers is 20 and the difference between them is 10. What are the numbers?

Solution

Let the two numbers be x and y.

Given that:

\( \begin{align*} x+y&=20 \rightarrow 1   \\ x-y&=10 \rightarrow 2 \end{align*} \) 

Consider second equation and find the value of x, i.e., 

\[\begin{align*} x-y&=10\\ x&=10+y\end{align*}\]

Substuting the value of x in the first equation to find the value of y.

\((10+y)+y=20\)

Simplify the equation in terms of y and find the value of y.

\[\begin{align*}10+2y&=20\\ 2y&=20-10\\ y&=\dfrac{10}{2}\\ y&=5\end{align*}\]

Substituting the value of y in Eq. first, we have,

\[\begin{align*}x+5 &=20\\ x&=20-5\\ x&= 15\end{align*}\]

\(\therefore\) Two numbers are 15 and 5.
Example 3

 

 

Jonathan has a box, the total number of $1 coins and $2 coins is 20 in that box. If the total coins amount to $46, find the number of coins of each denomination.

system of equations substitution method

Solution

Let the number of $1coins and $2 coins be x and y respectively.

Given that:

\( \begin{align*} x+y &=20 \rightarrow 1 \\ 2x+5y&=46 \rightarrow 2 \end{align*}\)

Consider first equation and find the value of x.

\[\begin{align*} x+y&=20 \\  x&=20-y\end{align*}\]

Substuting the value of x in the second equation to find the value of y.

\( 2(20-y)+5y=46\)

Simplify the equation in terms of y and find the value of y.

\[\begin{align*}40-2y+5y&=46\\ 3y &=46-40\\ y&=\dfrac{6}{3}\\ y&=2\end{align*}\]

Substituting the value of y in Eq. first, we have,

\[\begin{align*}x+2&=20\\ x&=20-2\\ x&= 18\end{align*}\]

\(\therefore\) Number of $1 coins =18

  Number of $2 coins = 2

 
Challenge your math skills
Challenging Questions
  1. Solve the equation by Elimination by substitution if \(6m-3n=32 \text{ and } x+y=-1\)

  2. The sum of the successors of the two numbers is 36 and the difference of their predecessors is 8. What are the numbers?

Interactive Questions 

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
Thinking out of the box
Think Tank
  1. Can you name all the five ways in which a set of two-equation in two variables can be solved?
  2. How many solutions are there for the equation in two variables?

Let's Summarize

The mini-lesson targeted the fascinating concept of the substitution method. The math journey around the substitution method starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

We hope you enjoyed learning about the substitution method with the simulations and practice questions. Now, you will be able to easily solve problems on substitution method, systems of equations substitution method, substitution method calculator, substitution method algebra.

Frequently Asked Questions (FAQs)

1. What is the substitution method?

 In the substitution method, the value of any one variable from any equation is substituted in the other equation.

2. How do you solve by substitution?

The method of solving "by substitution" involves:

  • Finding the value of any one variable from any given equation.
  • Substitute this value of the variable into the other equation.
  • simplify the equation thus obtained. to calculate the final value of the second variable.
  • Resubstitute the value into the original equation and hence find the final value of the first variable.

3. Write down the steps involved in the substitution method?

The method of solving "by substitution" involves:

  • Finding the value of any one variable from any given equation. 
  • Substitute this value of the variable into the other equation.
  • simplify the equation thus obtained. to calculate the final value of the second variable.
  • Resubstitute the value into the original equation and hence find the final value of the first variable.

4. How do you solve two equations using substitution?

The method to solve the two equations using substitution by solving one equation for one of the variables. Substitute the value of one variable into the other equation to find the other variable and solve. Resubstitute the value into the original equation.

5. Can the substitution method be used to solve the system of equations in three variables?   

Yes, the substitution method can be used to solve the system of equations in three variables.

6. What is the benefit of the substitution method?

The benefit of the substitution method is that it gives the exact values for the variables in the equations.

7. How do you solve substitution and elimination?

Substitution method
In the substitution method, the value of any one variable from anyone equation is substituted in the other equation.

Elimination method
In the elimination method, we either add or subtract the equations to get the solution.

8. How many solutions are there for linear equations in two variables?

There are three solutions for linear equations in two variables.

There are zero solutions. There is one solution. There are infinitely many solutions.

9. What is a two-variable equation?

A linear system of two equations with two variables is called a two-variable equation.

Download Pair of Linear Equations in 2 Variables Worksheets
Pair of Linear Equations in 2 Variables
grade 10 | Answers Set 1
Pair of Linear Equations in 2 Variables
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Pair of Linear Equations in 2 Variables
grade 10 | Answers Set 2
Pair of Linear Equations in 2 Variables
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