101 LearnersLast updated on August 27th, 2025
Substitution Method
The substitution method is one of the methods used for solving a system of linear equations in two variables. It is useful when one of the equations is already solved, or can be rearranged for one variable in terms of the other.
What is a Substitution Method?
The substitution method in algebra solves a system of equations having two or more variables. First, it isolates one variable in one of the equations, which forms a new expression. This expression is then substituted into the other equation. Doing so helps eliminate the other variable and solve the remaining equation with only one variable. Once solved, the value found is substituted back into the original equations to find the other variable. The answers can be verified by substituting the values of the variables in the original equations.
Difference Between Elimination and Substitution Method:
| Aspects | Elimination method | Substitution method |
| Meaning | Add or subtract the equation to cut out the one variable | Replace the one variable with the equivalent from another variable. |
| When to use | When variables have the same or opposite coefficients. | When one equation is already solved or easy to solve for a variable |
| Efficiency | Most efficient when equations have matching or opposite coefficients for one variable, making elimination easier. | Simple systems or when isolation is easy |
| Possible risk | Might lead to tricky fractions that are harder to simplify. | Substituting increases the number of steps, making it more prone to human errors. |
| Adaptability | Useful when the equations are written in different styles or formats. | Useful when equations are in standard form. |
What are the Methods of Substitution?
The substitution method expresses one variable in terms of the other. To do so, the first thing done is algebraic isolation of one of the variables. Let’s say we have an equation having 2 variables x and y. The method of substitution includes algebraic isolation, substitution, solving a single variable equation, back substitution, and verification. Given below are three steps explaining this method.
- First, solve one equation for one of the variables.
- Second, substitute that expression into the other equation and solve for the remaining variable.
- Third, substitute the found value back into either original equation to find the other variable.
Example:
Solve the equation using substitution method
x = 5 - 2y (1)
3x + y = 4 (2)
Solution
First:
Substitute equation (1) into equation (2). The equation gives x, substitute it into equation (2)
x = 5 - 2y
3x + y = 4
3(5 - 2y) + y = 4
Second, solve the equation for y
Multiply by 3 both inside the brackets
15 - 6y + y = 4
Combine the terms -6y and y we get 5y
15 - 5y = 4
Then change the coefficient
-5y = 4 - 15
-5y = - 11
y = 11/5
Third substitute y back into equation (1)
Now y = 11/5 in equation (1)
X = 5 - 2y
X = 5 - 2(11/5)
Then multiply
X = 5 - 22/5
Write 5 as a fraction with denominator 5
X = 25/5 - 22/5 = 3/5
The value of x and y is, x = 3/5, y = 11/5
Real-Life Applications of Substitution Methods:
We use substitution in everyday problems by swapping one unknown with an expression, so we can solve for both values more easily.
Problem-solving:
You have two choices—spending time with family or friends—and since they’re connected, you can use the substitution method to find a balance between them.
Business Planning:
When producing two related products, you can use the substitution method to figure out how many of each to make or sell to reach a target.
Planning for Trip:
While planning a trip with two parts—like driving and taking a train—if you know the speed and time of one part compared to the other, you can use substitution to figure out the total distance or time.
Time Management:
If you're dividing your time between two tasks—like studying and playing—and you know how much longer one takes than the other, you can use substitution to figure out how to split your time within the total hours you have.
Construction:
In construction, if you're using standard and special bricks and know the total number needed, you can use substitution to figure out how many of each to use, especially if standard bricks are cheaper, and you want to use more of them.
Common Mistakes and How to Avoid Them in Substitution Method
The substitution method, while helpful, can sometimes be confusing. Being aware of commonly occurring mistakes alerts students to keep them in mind and avoid them.
Mistake 1
Substituting the Wrong Equation.
Students may substitute the expression back into the same equation from which they isolated the variable, resulting in incorrect solutions.
For example, a given system;
x = 3y - 2 (1)
2x + y = 10 (2)
Students may use equation 1 to find the value of x and substitute the value in the same equation. This is wrong. Instead, after finding a substitute in equation 2.
Mistake 2
Substituting Without Brackets, Causing Sign Errors or Incorrect Simplification.
Use brackets around the expression you plug in to avoid mistakes.
Mistake 3
Confusing or Skipping Steps While Working with Algebraic Expressions or Equations.
Write and review every calculation clearly, and look for sign errors carefully.
Mistake 4
Solving for the Harder Variable First, Even When an Easier One is Available.
Choose the variable with a coefficient of 1 or -1instead of a large coefficient to avoid unnecessary complications.
Mistake 5
Assuming the Work is Done Without Confirming the Result.
Verify your answers by testing them in the original equations.
Solved Examples of Substitution Methods
Problem 1
Solve the equation using substitution methods: x = 3y + 2, 2x + y + 12
x is 387, and y is 87
Explanation
Substitute x from the first equation into the second
2(3y + 2) + y = 12
Multiply by 2 both inside the brackets
6y + 4 + y = 12
Combine the terms 6y and y we get y
7y + 4 = 12
Then change the coefficient
7y = 12 - 4
7y = 8
y = 8/7
Then find x
x = 3y + 2
x = 3(8/7) + 2
x = 24/7 + 2
Write 7 as a fraction with a denominator of 7
x = 24/7 + 14/7 = 38/7
The value of x is 38/7, and y is 8/7
Problem 2
Solve the equation using substitution methods: 2x − y = 3, x = y + 4
x = −1, y = −5
Explanation
Substitute x:
2(y+4) − y = 3
2y + 8 − y = 3
y + 8 = 3
y = 3 − 8 = −5
Find x:
x = −5 + 4 = −1
Answer is x = −1, y = −5
Problem 3
Solve the equation using substitution methods: x = y + 2, 2x + y = 13
x = 5, y = 3
Explanation
Substitute x into the second equation
2(y+2) + y = 13
Multiply by 2 both inside the brackets
2y + 4 + y = 13
Combining the terms 2y and y, we get y
3y + 4 = 13
Then change the coefficient and subtract 4:
3y = 9
Divide:
y = 3
Find x:
x = y + 2
x = 3 + 2 = 5
x = 5
The value of x is 5 and y is 3
Problem 4
Solve the equation using substitution methods: x=4y, x+y=10
x = 8, y = 2
Explanation
Substitute x = 4y into the second equation:
4y + y = 10
Combine the terms 5y and y we get y
5y =10
Divide:
y = 2
Find x:
x = 4 × 2 = 8
x = 8
Problem 5
Solve the equation using substitution methods: x = y + 4, 2x − y = 10
x = 6, y = 2
Explanation
Substitute x = y + 4:
Multiply by 2 both inside the brackets
2(y+4) − y = 10
Expand:
2y + 8 − y = 10
Then change the coefficient and subtract 8
y + 8 = 10
y = 2
Find x:
x = 2 + 4 = 6
x = 6
FAQs on Substitution Methods
1.Can substitution be used for word problems?
2.What is the substitution method in math?
3.When should I use the substitution method?
4.Why do I need parentheses when substituting?
5.How do I check if my solution is correct?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Substitution Method?
8.How do technology and digital tools in India support learning Algebra and Substitution Method?
9.Does learning Algebra support future career opportunities for students in India?
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