3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

You must have come across situations like the one given below : Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs 3, and a game of Hoopla costs 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20.

This introduction sets the stage for understanding how real-life situations can be modeled using mathematics. In this case, Akhila's experience at a fair is used to illustrate how we can form linear equations based on given data: the costs of rides and games and the limitations of her budget. By defining variables for rides and games, we can establish a mathematical way to analyze her choices.

Consider a scenario where you have $50 to spend on snacks and drinks at a party. If snacks cost$5 each and drinks cost $3 each, you can set up a similar linear equation to determine how many snacks and drinks you could buy within your budget.

Let us try this approach. Denote the number of rides that Akhila had by x, and the number of times she played Hoopla by y. Now the situation can be represented by the two equations:

1. y = x/2 (1)
2. 3x + 4y = 20 (2)

In this chunk, we define variables to represent the quantities of interest: 'x' for the number of Giant Wheel rides and 'y' for the number of times Hoopla was played. The first equation, y = x/2, describes the relationship between the two variables directly—Akhila played Hoopla half as many times as she rode the Giant Wheel. The second equation, 3x + 4y = 20, represents her spending constraint where the costs of rides and games total to $20.

Imagine you want to buy fruit and vegetables. If you buy 'x' pounds of apples at $2 per pound and 'y' pounds of carrots at$1 per pound, you can create a budgeting equation based on how much you can spend in total. This same idea translates directly to Akhila's problem.

A pair of linear equations which has no solution, is called an inconsistent pair of linear equations. A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations. A pair of linear equations which are equivalent has infinitely many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables. Note that a dependent pair of linear equations is always consistent.

This section introduces terms that categorize pairs of linear equations based on their solutions. An inconsistent pair implies that the lines representing the equations never meet, leading to no solutions. A consistent pair indicates they intersect at a single point, providing a unique solution. A dependent pair of equations has overlapping lines, meaning there are infinitely many solutions, as every point along the line satisfies both equations.

Think of inconsistent equations like two friends trying to meet at specific points in a park but arriving at different locations. A consistent equation would be when they agree to meet at one specific bench, while a dependent situation would be like walking together along a path where they both share every location.

1. the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).
2. the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations).
3. the lines may be coincident. In this case, the equations have infinitely many solutions [dependent (consistent) pair of equations].

This section describes the graphical representation of pairs of linear equations. When graphed, if two lines intersect at one point, it means there is a unique solution (consistent). If lines are parallel, they never meet, indicating no solutions exist (inconsistent). If they are coincident, every point on one line lies on the other line, thus producing infinitely many solutions (dependent). This visual element is crucial in understanding how these equations relate to one another.

Consider a scenario where two roads cross (intersect) at one point, two roads run side by side forever (parallel), or two roads lie perfectly on top of one another (coincident). Understanding this relationship helps visualize how different equations can behave in relation to each other.

Consider the following three pairs of equations. (i) x – 2y = 0 and 3x + 4y – 20 = 0 (The lines intersect) (ii) 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 (The lines coincide) (iii) x + 2y – 4 = 0 and 2x + 4y – 12 = 0 (The lines are parallel)

In this example, three pairs of linear equations demonstrate the three types of relationships: intersecting, coincident, and parallel. The first pair's lines intersect at one point, indicating a unique solution. The second pair's lines coincide, meaning there are infinitely many solutions. The third pair's lines are parallel, implying there are no solutions. Each scenario helps students visualize how linear equations can interact in different ways leading to various outcomes.

If we think of a scenario involving two friends trying to decide where to meet based on their different paths, the intersecting lines would represent the perfect meeting point, coincident lines would imply they are always together, and parallel lines would signify they can never share the same spot.

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