Methods to Solve a Pair of Linear Equations in Two Variables with Stepwise Examples
The concept of Pair of Linear Equations in Two Variables plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're preparing for Class 10, Olympiads, or just want to strengthen your Maths foundation, mastering this topic helps solve many day-to-day and exam-based problems.
What Is Pair of Linear Equations in Two Variables?
A Pair of Linear Equations in Two Variables is a set of two equations, each of the first degree, involving two unknowns (typically x and y). These equations represent straight lines when plotted on a graph. The solution to the pair is the set of values (x, y) that satisfy both equations at the same time. You’ll find this concept applied in areas such as simultaneous equations, graphical representation, and solving word problems based on ages, money, speed, and more.
Key Formula for Pair of Linear Equations in Two Variables
Here’s the standard formula:
\( \begin{cases}
a_1x + b_1y + c_1 = 0\\
a_2x + b_2y + c_2 = 0
\end{cases} \)
where \( a_1, a_2, b_1, b_2, c_1, c_2 \) are real numbers and \( x, y \) are variables. The aim is to find a common solution (x, y) that satisfies both equations simultaneously.
Standard Forms & Important Terms
| Term | Meaning | Example |
|---|---|---|
| Variable | Unknown to be found (usually x, y) | x, y |
| Coefficient | Number multiplied to a variable | 3, -5 in 3x − 5y |
| Constant | Stand-alone number in the equation | 6 in x + 2y = 6 |
| Ordered Pair | Solution of the pair as (x, y) | (2, 3) |
Cross-Disciplinary Usage
Pair of Linear Equations in Two Variables is not only useful in Maths but also plays an important role in Physics, Computer Science, and logical reasoning. For instance, finding the intersection of two paths, mixing solutions, or balancing costs all use this principle. Students preparing for CBSE Class 10, JEE, or NTSE will see its relevance in various questions.
How to Solve a Pair of Linear Equations in Two Variables
There are different methods to solve a pair of linear equations:
| Method | Steps | Best For |
|---|---|---|
| Graphical Method | Plot both equations on a graph; their intersection is the solution | Understanding concept visually |
| Substitution Method | Solve one equation for one variable, substitute into the next | Simple equations, fractions avoided |
| Elimination Method | Add/subtract equations to eliminate one variable, solve the other | Exam problems, stepwise calculation |
| Cross Multiplication | Apply the formula for both variables (for ax + by + c = 0 type) | Speed in objective exams |
Step-by-Step Illustration (Elimination Method Example)
Let’s solve:
\( 2x + 3y = 13 \) and \( x - 2y = -4 \)
1. Multiply the second equation by 2: \( 2x - 4y = -8 \)2. Subtract this from first equation:
[ \( 2x + 3y = 13 \) ]
[ \( 2x - 4y = -8 \) ]
Subtract:
(2x + 3y) - (2x - 4y) = 13 - (-8)
Gives: 7y = 21
3. So, y = 3.
4. Substitute y = 3 into the original equation:
x - 2×3 = -4 ⇒ x - 6 = -4 ⇒ x = 2.
Solution: (2, 3)
Speed Trick or Vedic Shortcut
When both equations are in simple standard form, you can use a cross multiplication trick:
- Arrange equations as: \( a_1x + b_1y + c_1 = 0 \), \( a_2x + b_2y + c_2 = 0 \)
- Apply cross-multiplication for x and y:
\( x = \frac{(b_1c_2 - b_2c_1)}{(a_1b_2 - a_2b_1)} \) \( y = \frac{(c_1a_2 - c_2a_1)}{(a_1b_2 - a_2b_1)} \)
This method saves time in competitive exams like NTSE and JEE. For more such tricks, Vedantu’s linear equations page (and live sessions) include plenty of speed-building shortcuts.
Try These Yourself
- Draw the graphs of \( x + y = 4 \) and \( 2x - y = 1 \). Where do they meet?
- Solve \( x + 3y = 7 \) and \( 2x - y = 5 \) by the substitution method.
- Check if \( x = 3, y = 2 \) satisfies \( 3x + 2y = 13 \) and \( x - y = 1 \).
- Identify if the equations \( 2x + 3y = 5 \) and \( 4x + 6y = 10 \) are consistent or dependent.
Frequent Errors and Misunderstandings
- Mixing up coefficients (double-check before adding/subtracting equations!)
- Not rearranging equations correctly before applying a method
- Forgetting to check solutions in both equations
- Graphing errors: misreading scales or axes
Relation to Other Concepts
The idea of Pair of Linear Equations in Two Variables connects closely with topics such as Linear Equations in One Variable and Simultaneous Equations. Mastering this topic helps build confidence for algebra, coordinate geometry, and higher-level Maths such as systems of equations and matrices.
Classroom Tip
A quick way to visually remember the solution is: if two lines intersect, there’s one solution; if they’re parallel, no solution; and if they overlap (coincide), infinite solutions! Vedantu’s teachers recommend color-coding equations or using summary tables during revision to make this easy for mobile users and during last-minute prep.
Wrapping It All Up
We explored Pair of Linear Equations in Two Variables — from definition, formula, working examples, shortcuts, common mistakes, and connections to other subjects. Keep practicing problems from NCERT and worksheets or try out real-life puzzles. To deepen your understanding, explore more resources and live sessions at Vedantu for doubt clearance and extra practice.
Explore More at Vedantu
- Linear Equations in One Variable – Great for Class 8–9 foundation
- Elimination Method – Master this favored CBSE technique
- Cross-Multiplication Method – For quick MCQ solutions
FAQs on Pair of Linear Equations in Two Variables: Complete Guide
1. What is a pair of linear equations in two variables?
A pair of linear equations in two variables consists of two equations, each having two variables (usually denoted as x and y) raised to the power of one. The general form is: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, where a₁, b₁, c₁, a₂, b₂, and c₂ are constants.
2. What are the common methods to solve a pair of linear equations?
The most common methods are:
- Graphical Method: Plotting both equations on a graph and finding the point of intersection (if it exists).
- Substitution Method: Solving one equation for one variable and substituting that expression into the other equation.
- Elimination Method: Manipulating the equations to eliminate one variable and then solving for the remaining variable.
- Cross-multiplication Method: A direct formula-based method to quickly find the solution.
3. Can you solve a pair of linear equations using a graph?
Yes. The graphical method involves plotting both linear equations on a coordinate plane. The point where the two lines intersect represents the solution (the values of x and y that satisfy both equations). If the lines are parallel, there's no solution; if they coincide, there are infinitely many solutions.
4. How do you apply the elimination method?
The elimination method involves making the coefficients of either x or y opposites in the two equations. Then, add the equations together to eliminate that variable. Solve the resulting equation for the remaining variable. Finally, substitute this value back into either of the original equations to find the value of the other variable.
5. What are some real-life examples of a pair of linear equations?
Real-world applications include problems involving:
- Mixture problems: Determining the amounts of two different solutions needed to create a desired mixture.
- Speed and distance: Finding the speeds and times of two objects moving towards or away from each other.
- Cost and quantity: Calculating the cost and number of items purchased from two different stores.
6. How do I use the substitution method to solve a pair of linear equations?
In the substitution method, solve one equation for one variable (e.g., solve for x in terms of y). Substitute this expression into the second equation, which will now only have one variable. Solve for that variable, and then substitute the value back into the first equation to find the value of the other variable.
7. What is the cross-multiplication method for solving linear equations?
The cross-multiplication method is a shortcut for solving pairs of linear equations. It uses a specific formula to directly calculate the values of x and y without the need for elimination or substitution steps.
8. What does it mean if a pair of linear equations has no solution?
If a pair of linear equations has no solution, it means the lines represented by the equations are parallel and never intersect. This occurs when the coefficients of x and y are proportional, but the constant terms are not proportional (a₁/a₂ = b₁/b₂ ≠ c₁/c₂).
9. What does it mean if a pair of linear equations has infinitely many solutions?
A pair of linear equations has infinitely many solutions if the lines representing the equations coincide (they are the same line). This happens when the ratios of the coefficients and constant terms are all equal (a₁/a₂ = b₁/b₂ = c₁/c₂).
10. How can I check if my solution is correct for a pair of linear equations?
Substitute the values you found for x and y back into both original equations. If both equations are true with those values, your solution is correct. If not, there's been a mistake in the solving process.
11. What are consistent and inconsistent pairs of linear equations?
A consistent pair of linear equations has at least one solution (either one unique solution or infinitely many). An inconsistent pair has no solution because the lines are parallel. Consistency can be determined by comparing the ratios of coefficients (a₁/a₂, b₁/b₂, and c₁/c₂).
