Q) Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair: x – 5 y = 6; 2 x – 10 y = 12.

(Q 33 – 30/2/2 – CBSE 2026 Question Paper)

Ans:

We are given following two equations: 
1. x – 5 y = 6
2. 2 x – 10 y = 12

(i) Graphical representation:

Step 1: To plot on graph, let’s calculate point where these lines, cut x-axis and y-axis:

1. For equation 1: x – 5 y = 6:

∵ on Y-axis, x = 0; ∴ for x = 0, y = –

Line will pass through (0, – )

and ∵ on X-axis, y = 0;  ∴ y = 0, x = 6

Line will pass through (6, 0)

2. Now, for equation: 2 x – 10 y = 12

for x = 0, y = –

Line will pass through (0, – )

for y = 0, x = = 6

Line will pass through (6, 0)

Step 2: Next, we plot these 2 equations on graph.

We can see that both equations are identical and they represent the same straight line.

(ii) Comment on Consistency:

[Note: Before commenting on “on the condition of consistency of the given pair of linear equations”; let’s first understand the consistency & dependency conditions of linear equations:
A. A system of linear equations is “CONSISTENT” if it has at least one solution. This includes two cases:
    (i)  Unique solution: The two lines intersect at exactly one point.                             In such pair,
    (ii) Infinite solutions: The two lines coincide i.e. lie on top of each other. It means that each point on a line is a solution, as it lies on other line as well. Such a pair of linear equations is also “DEPENDENT“, where one equation is a continuous multiple of the other, representing the same line. This leads to infinitely many solutions. In such pair,
B. A system of linear equations is called “INCONSISTENT“, if it has no solution. Such lines are parallel lines and do not intersect at any point.
In such pair, ]

Step 3: We are given pair of linear equations as:

1. x – 5 y = 6

∴ x – 5 y – 6 = 0

and 2 x – 10 y = 12 or

∴ 2 x – 10 y – 12 = 0

Let’s compare our given pair of equations with standard pair of linear equations: a₁ x + b₁ y + c₁ = 0 and a₂ x + b₂ y + c₂ = 0

We have a₁ = 1, b₁ = – 5, c₁= – 6 and a₂ = 2, b₂ = – 10, c₂ = – 12

Step 4: By comparing ratios of coefficients, we get:

;

;

;

∵ For this pair of linear equations,

Therefore, these lines are consistent and dependent. Both lines will have infinitely many solutions.

Final Answer: Graphically, both equations represent the same line. and Condition-wise, both equations are consistent and dependent with infinitely many solutions.

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