How to Solve Linear Equations by the Substitution Method?
The concept of Substitution Method plays a key role in mathematics and is widely used to solve systems of equations, especially linear and algebraic problems in classes 9, 10, and other competitive exams. Understanding the substitution method makes equation solving simple and efficient.
What Is the Substitution Method?
The substitution method is defined as a stepwise process to solve a system of equations by expressing one variable in terms of another and then substituting that value in the other equation. You’ll find this concept applied in algebraic equations, linear systems of two variables, and even some quadratic and word problems.
Key Formula for Substitution Method
Here’s the standard formula used in the substitution method: Find the value of one variable (e.g.,
Cross-Disciplinary Usage
The substitution method is not only a Maths concept but is also relevant in Physics (for solving motion equations), Computer Science (for algorithm logic), and logical reasoning in daily life. Students preparing for Olympiads, NTSE, or board exams will encounter substitution method questions frequently, including in the Algebraic Equations and Linear Equations chapters.
Step-by-Step Illustration
Let's see a solved example using the substitution method:
| Step | Description |
|---|---|
| 1 | Start with the system: |
| 2 | Isolate x in (i): |
| 3 | Substitute |
| 4 | Solve for y: |
| 5 | Back substitute to find x: |
| 6 | Final answer: (x, y) = (5, 2) |
Speed Trick or Vedic Shortcut
Substitution method works fast when you notice a variable already isolated or with a coefficient of 1. Always pick such variables to minimize calculation. For competitive exams, quickly plug values using cross-multiplication if one variable is easily expressed from one equation.
Example Trick: If one equation is already in
Try These Yourself
- Solve by substitution:
- Find x, y using substitution:
- If
Frequent Errors and Misunderstandings
- Substituting back into the same equation (always substitute into the other equation).
- Missing negative signs or arithmetic errors while isolating variables.
- Forgetting to verify the final answer in both original equations.
Relation to Other Concepts
The substitution method relates closely to the Elimination Method for solving simultaneous equations. Mastery of substitution also helps with solving Word Problems and graphical solutions involving equations of a line.
Classroom Tip
A quick way to remember the substitution method: ISOLATE — SUBSTITUTE — SOLVE — VERIFY. Vedantu’s live class teachers often use colorful arrows and circle the isolated variable in steps to make visual learning clearer for younger students.
Wrapping It All Up
We explored the substitution method—from its definition to formula, step-by-step solved examples, mistakes to avoid, and its connection with related algebra concepts. Continue practicing with Vedantu for more confidence in solving equations by substitution, and use the online Substitution Method Calculator for quick answers!
Related Links for More Practice
FAQs on Substitution Method in Maths: Step-by-Step Guide and Examples
1. What is the substitution method in Maths?
The substitution method is an algebraic technique used to solve systems of equations, typically linear equations with two variables. It involves solving one equation for one variable, then substituting that expression into the other equation to solve for the remaining variable. This process allows you to find the values that satisfy both equations simultaneously.
2. How do you solve equations using the substitution method?
Solving equations using the substitution method involves these steps:
1. **Isolate a variable:** Solve one equation for one variable (e.g., solve for 'x' in terms of 'y').
2. **Substitute:** Substitute the expression you found in Step 1 into the other equation.
3. **Solve:** Solve the resulting equation for the remaining variable.
4. **Back-substitute:** Substitute the value you found back into the expression from Step 1 (or any original equation) to find the value of the other variable.
5. **Verify:** Check your solution by plugging the values into both original equations to ensure they satisfy both equations.
3. Can you provide a simple example of the substitution method?
Let's solve: x + y = 5 and 2x - y = 4.
1. From the first equation, solve for x: x = 5 - y.
2. Substitute this into the second equation: 2(5 - y) - y = 4.
3. Solve for y: 10 - 2y - y = 4 => -3y = -6 => y = 2.
4. Substitute y = 2 back into x = 5 - y: x = 5 - 2 = 3.
5. Solution: (x, y) = (3, 2).
4. What are the advantages of using the substitution method?
The substitution method is particularly useful when:
• One variable is already isolated in one of the equations.
• One equation can easily be solved for one variable without introducing fractions.
• Solving systems involving non-linear equations (though more complex).
5. What are the limitations of the substitution method?
The substitution method can become cumbersome if both equations have complex coefficients or if solving for one variable results in fractional expressions.
6. How is the substitution method different from the elimination method?
Both solve systems of equations, but the substitution method replaces a variable with an equivalent expression, while the elimination method adds or subtracts equations to eliminate a variable.
7. When is it best to use the substitution method over the elimination method?
Use substitution when one variable is already isolated or easily isolated, or when dealing with non-linear equations. Use elimination when coefficients of variables are simple and easily manipulated to cancel each other.
8. Can the substitution method be used for non-linear equations?
Yes, the substitution method can be applied to non-linear systems, but the process is generally more complex. It often requires solving a quadratic or higher-degree equation for one of the variables.
9. What are common mistakes to avoid when using the substitution method?
Common mistakes include incorrect substitution, errors in solving the resulting equation (especially with signs), and forgetting to back-substitute to find the second variable.
10. How can I check my answer when using the substitution method?
Always verify your solution by plugging the values back into both original equations. If both equations are satisfied, your solution is correct.
11. Are there any online tools or calculators that can help with the substitution method?
Yes, many online systems of equations solvers can help check your work and provide step-by-step solutions. Searching for "substitution method calculator" will yield several results.
12. Why does the substitution method work mathematically?
The substitution method relies on the transitive property of equality: if a = b and b = c, then a = c. When you substitute an expression for a variable, you're replacing it with an equivalent value, maintaining the equality and allowing you to solve the system.
