Pair of Linear Equations (Concept) MCQs Quiz | Class 10

Understanding Pair of Linear Equations

A pair of linear equations in two variables is a fundamental concept in Class 10 Mathematics, forming the basis for understanding various real-world problems. This section aims to consolidate your understanding of their meaning, standard forms, and different methods of representation.

1. Meaning of a Linear Equation in Two Variables

A linear equation in two variables, say x and y, is an equation that can be written in the form `ax + by + c = 0`, where a, b, and c are real numbers, and a and b are not both zero. The term ‘linear’ comes from the fact that its graph is always a straight line. When we talk about a ‘pair’ of linear equations, we are referring to two such equations considered together.

For example, 2x + 3y = 7 and 5x - 2y = 8 form a pair of linear equations. The solution to a pair of linear equations is a pair of values (x, y) that satisfies both equations simultaneously. Geometrically, this solution corresponds to the point of intersection of the two lines represented by the equations.

2. Standard Forms of a Pair of Linear Equations

The general or standard form for a pair of linear equations in two variables x and y is:

• a1x + b1y + c1 = 0
• a2x + b2y + c2 = 0

Here, a1, b1, c1, a2, b2, and c2 are real numbers. It is important that a1^2 + b1^2 is not equal to zero and a2^2 + b2^2 is not equal to zero, ensuring that both equations are indeed linear in two variables.

For instance, in the equation 3x - 4y = 10, we can write it in standard form as 3x - 4y - 10 = 0, where a1=3, b1=-4, c1=-10.

3. Representation of a Pair of Linear Equations

A pair of linear equations can be represented in two primary ways:

Graphical Representation:

When each linear equation in a pair is plotted on a graph, it forms a straight line. The nature of the lines formed determines the number of solutions:

• Intersecting Lines: If the lines intersect at a single point, the pair of equations has a unique solution. This system is consistent. (Condition: a1/a2 is not equal to b1/b2)
• Parallel Lines: If the lines are parallel and never intersect, the pair of equations has no solution. This system is inconsistent. (Condition: a1/a2 = b1/b2 is not equal to c1/c2)
• Coincident Lines: If the lines overlap completely (one line lies exactly on the other), the pair of equations has infinitely many solutions. This system is consistent and dependent. (Condition: a1/a2 = b1/b2 = c1/c2)

Algebraic Representation:

Algebraic methods are used to find the exact numerical values of x and y that satisfy both equations. The common methods include:

• Substitution Method: Express one variable in terms of the other from one equation and substitute it into the second equation.
• Elimination Method: Multiply one or both equations by suitable non-zero constants so that the coefficients of one variable become numerically equal, then add or subtract the equations to eliminate that variable.
• Cross-Multiplication Method: A specific formula-based method derived from elimination for solving the equations directly.

Summary of Conditions for Solvability

Quick Revision Points:

• A linear equation in two variables represents a straight line on a graph.
• A pair of linear equations signifies two lines.
• Solutions are points that lie on both lines.
• ax + by + c = 0 is the standard form.
• Graphical method visually identifies the number of solutions.
• Algebraic methods (substitution, elimination, cross-multiplication) find exact solutions.

Practice Questions:

1. Write the standard form of a linear equation in two variables.
2. Give an example of a pair of linear equations that has no solution.
3. For what type of lines does a pair of linear equations have infinitely many solutions?
4. If the ratio a1/a2 is not equal to b1/b2, what kind of solution does the pair of equations have?
5. The equation 2x + 0y = 5 is a linear equation in two variables. True or False?