3.2. Graphical Method of Solution of a Pair of Linear Equations

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.

In linear equations, we can categorize our equations into three types based on their solutions. An inconsistent pair indicates no solution exists, meaning the lines representing the equations do not intersect at any point. A consistent pair means there is at least one solution, typically being two lines that intersect at a single point. Lastly, dependent pairs imply that the two lines overlap entirely, giving us many solutions as every point along the line is valid.

Imagine a scenario where two people are looking for common ground to meet. If one person states they can only be at location A, and the other states they will only be at location B (which is far away), it’s like our inconsistent pairs, where meeting is impossible. However, if both can meet at one park (the solution), that’s akin to a consistent pair. Finally, if both agree to meet anywhere along the same path (dependent pair), that means we have countless meeting points.

We can now summarise the behaviour of lines representing a pair of linear equations in two variables and the existence of solutions as follows:

(i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).

(ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations).

(iii) the lines may be coincident. In this case, the equations have infinitely many solutions [dependent (consistent) pair of equations].

When we graph linear equations, we observe how the lines behave in relation to each other. If two lines cross at a point, that point represents the unique solution to the equations. If they run parallel, they will never meet, indicating no solutions exist. In contrast, if the lines lie on top of each other, they are equivalent, and any point on the line is a solution.

Think of a busy intersection: the crossing of the roads represents the unique solution as cars can only meet at that point. Parallel roads are like two friends who live on the same street but never cross paths, while shared lanes or paths indicate they can go anywhere together—every point along the way is an option.

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 these examples, we are looking at specific pairs of equations. The first pair of equations intersect, indicating a unique solution exists. The second pair represents coincident lines, meaning there are infinitely many solutions since every point on the line is a solution to both equations. Finally, the third pair is parallel and signifies that no solutions can be found as they will never meet.

Imagine three scenarios: in the first, two friends meet at a coffee shop (the intersecting lines). In the second scenario, twins are always together (coincident means every moment together can be shared). In the third, two friends on different subway lines—regardless of how often they travel, they will never meet (parallel lines)!

From the table above, you can observe that if the lines represented by the equation ax + by + c = 0 and ax + by + c = 0 are (i) intersecting, then a1 ≠ a2. (ii) coincident, then a1 = a2 = c1. (iii) parallel, then a1 = a2 ≠ c1.

The properties of the coefficients (a1, b1, c1) give insight into the relationships between the two lines. If the ratios are different for 'a', the lines intersect. If they are equal for 'a' and 'b' but have different 'c' values, the lines are parallel. Lastly, if all three ratios are equal, it indicates the lines are coincident.

Think of these coefficients as ingredients in a recipe. If you have the same ingredients in the same amount (coincident), you get the same dish (infinite solutions). If the amounts are different but the ingredients are the same (intersecting), you will create different dishes but share some parts. Lastly, if you use completely different ingredients (parallel), you won’t be able to make something similar at all.

Let us now consider some more examples to illustrate it.

We can use equations to demonstrate how to find solutions graphically. By plotting the equations on a coordinate plane, we can visually see where they intersect, indicating the solution to the pair of equations. If they don’t intersect, we can conclude there are no solutions, and if they coincide, we can state there are infinitely many solutions.

Imagine a treasure map with two different trails marked by two friends looking for treasure. If their paths cross at a single point, they found the treasure! If one friend’s trail overlaps with the other, then they will find treasure everywhere along that path. But if they are walking on two completely different paths with no intersection, they will never uncover the treasure together.

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