Ch 3.3 Maths Class 10

Chapter 3.3: Understanding and Applying Pair of Linear Equations in Two Variables (Class 10 Maths)

This article walks through Chapter 3.3 of Class 10 mathematics, focusing on pairs of linear equations in two variables. We will explore various methods for solving these equations, their graphical representation, and real-world applications. And understanding this chapter is crucial for building a strong foundation in algebra and its practical applications. We'll cover the key concepts with detailed explanations and examples, making it accessible for all learners.

Introduction: What are Pair of Linear Equations in Two Variables?

A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are the variables. A pair of linear equations means we have two such equations, simultaneously involving the same two variables. For example:

• 2x + 3y = 7
• x - y = 1

These equations represent straight lines when graphed on a Cartesian plane. That's why the solution to a pair of linear equations is the point (x, y) where these two lines intersect. This point satisfies both equations simultaneously.

Methods for Solving Pairs of Linear Equations

You've got several methods worth knowing here. We will explore the most common ones:

1. Graphical Method:

This method involves plotting the graphs of both equations on the same Cartesian plane. The point of intersection of the two lines represents the solution.

Steps:

1. Rearrange: Rewrite each equation in the form y = mx + c, where 'm' is the slope and 'c' is the y-intercept.
2. Plot: For each equation, find at least two points that satisfy the equation and plot them on the graph. Draw a straight line through these points.
3. Identify Intersection: The point where the two lines intersect is the solution (x, y).

Example:

Let's solve the equations graphically:

• 2x + y = 5
• x - y = 1

Solution:

First, rearrange the equations:

• y = -2x + 5
• y = x - 1

Now, plot these lines: For the first equation, if x = 0, y = 5, and if x = 1, y = 3. Think about it: plot these points and draw the lines. Day to day, you will find they intersect at the point (2, 1). For the second equation, if x = 0, y = -1, and if x = 1, y = 0. Which means, the solution is x = 2 and y = 1.

Limitations: The graphical method is not always accurate, particularly when the solution involves fractions or decimals, as precise plotting can be challenging.

2. Algebraic Methods:

Algebraic methods offer more precision. The most common are:

a) Substitution Method:

This method involves solving one equation for one variable in terms of the other, and then substituting this expression into the second equation.

Steps:

1. Solve for a variable: Solve one equation for one variable (e.g., solve for 'x' in terms of 'y' or vice versa).
2. Substitute: Substitute this expression into the other equation.
3. Solve: Solve the resulting equation for the remaining variable.
4. Back-substitute: Substitute the value obtained back into either of the original equations to find the value of the other variable.

Example:

Solve the system:

• x + y = 5
• x - y = 1

Solution:

1. Solve the first equation for x: x = 5 - y
2. Substitute this into the second equation: (5 - y) - y = 1
3. Solve for y: 5 - 2y = 1 => 2y = 4 => y = 2
4. Back-substitute y = 2 into x = 5 - y: x = 5 - 2 = 3

So, the solution is x = 3 and y = 2.

b) Elimination Method:

This method involves eliminating one variable by adding or subtracting the two equations.

Steps:

1. Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., one is +2x and the other is -2x).
2. Add or subtract: Add or subtract the two equations to eliminate the chosen variable.
3. Solve: Solve the resulting equation for the remaining variable.
4. Back-substitute: Substitute the value obtained back into either of the original equations to find the value of the other variable.

Example:

If you found this helpful, you might also enjoy word problems with rational expressions or who is slim in the book of mice and men.

Solve the system:

• 2x + y = 5
• x - y = 1

Solution:

1. The coefficients of 'y' are already opposites (+1 and -1).
2. Add the two equations: (2x + y) + (x - y) = 5 + 1 => 3x = 6 => x = 2
3. Substitute x = 2 into either equation (let's use the second): 2 - y = 1 => y = 1

That's why, the solution is x = 2 and y = 1.

c) Cross-Multiplication Method:

This is a shortcut method derived from the elimination method and particularly useful for solving equations quickly. It's applicable when the equations are in the standard form (ax + by + c = 0).

The formula is:

x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = -1 / (a1b2 - a2b1)

Where: a1, b1, c1 are coefficients of the first equation, and a2, b2, c2 are coefficients of the second equation.

Example:

Solve:

• 2x + 3y - 7 = 0
• x - y - 1 = 0

Solution:

Applying the formula:

x / ((3)(-1) - (-1)(-7)) = y / ((-7)(1) - (-1)(2)) = -1 / ((2)(-1) - (1)(3))

x / (-10) = y / (-5) = -1 / (-5)

x = 2, y = 1

Graphical Representation and Types of Solutions

Graphically, the solution to a pair of linear equations represents the point of intersection of the two lines. Even so, there are three possibilities:

1. Unique Solution: The lines intersect at exactly one point. This is the most common case, indicating a consistent and independent system of equations.

2. No Solution (Inconsistent): The lines are parallel and never intersect. This means the system of equations has no solution because there is no point that satisfies both equations simultaneously. Worth knowing.

3. Infinitely Many Solutions (Dependent): The lines are coincident (they overlap completely). This means any point on the line satisfies both equations, resulting in infinitely many solutions.

Real-World Applications

Pair of linear equations have numerous applications in real-world problems. Here are some examples:

• Mixture Problems: Determining the amounts of two different substances needed to create a desired mixture with specific properties.
• Speed and Distance Problems: Finding the speeds and times of two objects moving at different rates.
• Cost and Revenue Problems: Analyzing cost and revenue functions to determine break-even points.
• Age Problems: Solving problems involving the ages of individuals with given relationships.
• Number Problems: Finding two unknown numbers based on their sum or difference and other relationships.

Frequently Asked Questions (FAQ)

• Q: What if the equations are not in the standard form? A: You need to rearrange them into the standard form (ax + by + c = 0) before applying any of the methods.

• Q: Which method is the best? A: There's no single "best" method. The choice depends on the specific equations and your personal preference. The elimination method is often efficient for simple equations, while substitution works well when one variable is easily isolated. The graphical method is helpful for visualizing the solution but lacks precision. The cross-multiplication method is a quick way to get the solution but requires memorizing the formula.

• Q: How do I check my solution? A: Substitute the values of x and y obtained into both original equations. If both equations are satisfied, the solution is correct.

• Q: What does it mean if I get a solution like x = 0 and y = 0? A: This simply means the lines intersect at the origin (0, 0) of the Cartesian plane. It's a valid solution.

• Q: What if I get a contradictory statement while solving (e.g., 2 = 5)? A: This indicates that the system of equations is inconsistent, and there is no solution.

Conclusion

Understanding pairs of linear equations in two variables is fundamental to higher-level mathematics and its applications. Mastering the different methods for solving these equations, their graphical interpretations, and understanding the implications of unique, no, or infinitely many solutions are crucial skills for any student. Worth adding: practice is key to becoming proficient in solving these types of problems, and by applying the steps and techniques outlined in this article, you will be well-equipped to tackle any challenge related to Chapter 3. 3 of your Class 10 mathematics curriculum. Remember to visualize the problem, choose the most suitable method, and always check your solution. Good luck!