Class 10 Maths Chapter 3.3

Conquer Class 10 Maths Chapter 3.3: A Deep Dive into Linear Equations in Two Variables

Chapter 3.3 of Class 10 mathematics typically focuses on linear equations in two variables. This crucial chapter builds upon your foundational knowledge of algebra and introduces powerful techniques for solving problems involving two unknowns. Understanding this chapter is vital, not only for your upcoming exams but also for laying the groundwork for more advanced mathematical concepts in the future. This comprehensive guide will break down the core concepts, provide step-by-step solutions, and address frequently asked questions to ensure you master this topic.

Introduction to Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the standard form: ax + by + c = 0, where 'a', 'b', and 'c' are constants (real numbers), and 'x' and 'y' are the variables. The key characteristic is that the highest power of each variable is 1. This means there are no squared terms (x², y²), cubed terms (x³, y³), or any other higher powers. The graph of a linear equation in two variables is always a straight line.

Understanding the standard form is crucial because it allows you to easily identify the coefficients (a, b, c) which will be useful in various solving methods.

Graphical Representation of Linear Equations

Visualizing linear equations through their graphs provides a powerful understanding of their solutions. Every point (x, y) that lies on the line represents a solution to the equation. To graph a linear equation:

1. Find at least two points that satisfy the equation. You can do this by choosing a value for x, substituting it into the equation, and solving for y. Repeat this process with another x value. It's often easiest to choose x = 0 and solve for y, and then choose y = 0 and solve for x. These points will typically represent the x and y intercepts of the line.

2. Plot the points on a Cartesian plane (coordinate system).

3. Draw a straight line passing through these points. This line represents all the possible solutions to the linear equation.

Example: Let's graph the equation 2x + y - 4 = 0.

• If x = 0, then y = 4. So one point is (0, 4).
• If y = 0, then 2x = 4, so x = 2. The other point is (2, 0).
• Plot (0, 4) and (2, 0) on a graph and draw a straight line connecting them. This line is the graphical representation of the equation 2x + y - 4 = 0.

Solving Linear Equations in Two Variables Algebraically

While graphical methods offer a visual understanding, algebraic methods provide precise and efficient solutions. Several techniques exist, each suitable for different situations:

1. Elimination Method

The elimination method involves manipulating the equations to eliminate one variable, leaving you with an equation in a single variable which is easily solved. This is achieved by making the coefficients of one variable opposites.

Steps:

1. Multiply one or both equations by constants to make the coefficients of either x or y opposites (e.g., if one equation has 2x, and the other has -2x).

2. Add the two equations together. This will eliminate the variable with opposite coefficients.

3. Solve the resulting equation for the remaining variable.

4. Substitute the value obtained back into either of the original equations to solve for the other variable.

Example: Solve the system:

x + y = 5 x - y = 1

Adding the two equations eliminates 'y': 2x = 6, so x = 3. Substituting x = 3 into the first equation gives 3 + y = 5, so y = 2. The solution is x = 3, y = 2.

2. Substitution Method

The substitution 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 one equation for one variable (e.g., solve for x in terms of y or vice versa).

2. Substitute this expression into the other equation. This will create an equation with only one variable.

3. Solve the resulting equation.

4. Substitute the value obtained back into either of the original equations to find the value of the other variable.

Example: Solve the system:

x + 2y = 7 x - y = 1

From the second equation, x = y + 1. Substitute this into the first equation: (y + 1) + 2y = 7. This simplifies to 3y = 6, so y = 2. Substituting y = 2 back into x = y + 1 gives x = 3. The solution is x = 3, y = 2.

3. Cross-Multiplication Method

The cross-multiplication method is a shortcut for solving linear equations in two variables, particularly effective when the equations are in the standard form.

Steps:

For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

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

Example: Solve the system:

2x + 3y - 7 = 0 3x + 2y - 8 = 0

Applying the cross-multiplication method:

x / ((3)(-8) - (2)(-7)) = y / ((-7)(3) - (-8)(2)) = 1 / ((2)(2) - (3)(3))

x / (-24 + 14) = y / (-21 + 16) = 1 / (4 - 9)

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

x = 2 and y = 1

The solution is x = 2, y = 1.

Consistent and Inconsistent Systems

A consistent system of linear equations has at least one solution. This means the lines representing the equations either intersect at one point (unique solution) or coincide (infinite solutions). An inconsistent system has no solution; the lines representing the equations are parallel and never intersect.

You can determine the consistency of a system by comparing the ratios of the coefficients of the variables:

• Unique solution: a₁/a₂ ≠ b₁/b₂
• Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Applications of Linear Equations in Two Variables

Linear equations in two variables are widely applicable in various real-world scenarios, including:

• Mixture problems: Determining the quantities of different ingredients to create a specific mixture.
• Speed and distance problems: Calculating speeds and distances based on given information.
• Age problems: Solving problems related to the ages of individuals.
• Cost and profit problems: Analyzing cost and profit scenarios in business.

Frequently Asked Questions (FAQ)

• Q: What if I get a solution that doesn't satisfy both equations? A: Double-check your calculations in each step of the chosen method. A common error is an arithmetic mistake in substitution or elimination.

• Q: Which method is best for solving linear equations? A: The best method depends on the specific equations. The elimination method is often efficient when coefficients are easily manipulated. The substitution method works well when one variable is easily isolated. The cross-multiplication method offers a concise solution in standard form.

• Q: How do I know if I have solved the equations correctly? A: Substitute your solutions (x and y values) back into both original equations. If both equations are satisfied, your solution is correct.

• Q: What if the lines are parallel? A: Parallel lines indicate an inconsistent system with no solution. This occurs when the ratios of the coefficients of x and y are equal, but the ratio of the constants is different.

• Q: What if the lines coincide? A: Coinciding lines indicate a consistent system with infinite solutions. This happens when all the ratios of the coefficients (including constants) are equal.

Conclusion

Mastering Chapter 3.3 on linear equations in two variables is a cornerstone of your Class 10 mathematics journey. By understanding the various methods – graphical, elimination, substitution, and cross-multiplication – and their applications, you will be well-equipped to tackle diverse problems. Remember that practice is key; the more you solve problems, the more confident and proficient you will become. Don't hesitate to review the concepts, rework examples, and seek clarification if needed. With dedicated effort, you can conquer this chapter and build a strong foundation for future mathematical studies. Good luck!