Given the linear equation x-2y=6 write another linear equation in these two
variables, such that the geometrical representation of the pair so formed is:
(i) coincident lines (ii) intersection line
Answers
Answer:
1. The required equation is 2x - 4y - 12 = 0.
2. The required equation is x - 3y + 2 = 0.
Step-by-step-explanation:
We have given a linear equation in two variables.
We have to find other equation in the same variables so that the graph of both equations will be
1) coincident lines
2) intersecting lines
The given simultaneous equation is x - 2y = 6.
1.
To form a linear equation such that the graph of both equations will be a coincident line, the following condition must be satisfied.
For linear equations having coincident lines, the condition is -
a₁ / a₂ = b₁ / b₂ = c₁ / c₂
The standard form of the given equation is x - 2y - 6 = 0.
Here,
- a₁ = 1
- b₁ = - 2
- c₁ = - 6
Now,
Let the ratio of all the coefficients of both equations be 1 / 2.
∴ a₁ / a₂ = 1 / 2
⇒ 1 / a₂ = 1 / 2
⇒ a₂ = 1 × 2
∴ a₂ = 2
Now,
⇒ b₁ / b₂ = 1 / 2
⇒ - 2 / b₂ = 1 / 2
⇒ b₂ = - 2 × 2
⇒ b₂ = - 4
Now,
⇒ c₁ / c₂ = 1 / 2
⇒ - 6 / c₂ = 1 / 2
⇒ c₂ = - 6 × 2
⇒ c₂ = - 12
∴ a₂ = 2, b₂ = - 4, c₂ = - 12
∴ The required equation is 2x - 4y - 12 = 0.
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2.
To form a linear equation such that the graph of both equations will be a intersecting line, the following condition must be satisfied.
For linear equations having intersecting lines, the condition is -
a₁ / a₂ ≠ b₁ / b₂
We have from the given equation x - 2y - 6 = 0,
- a₁ = 1
- b₁ = - 2
- c₁ = - 6
Now,
Let the ratio of a₁ & a₂ be 1.
While the ratio of b₁ & b₂ be 2 / 3.
Now,
∴ a₁ / a₂ = 1 / 1
⇒ 1 / a₂ = 1
⇒ a₂ = 1 × 1
⇒ a₂ = 1
Now,
⇒ b₁ / b₂ = 2 / 3
⇒ - 2 / b₂ = 2 / 3
⇒ 2b₂ = - 2 × 3
⇒ 2b₂ = - 6
⇒ b₂ = - 6 / 2
⇒ b₂ = - 3
Let the constant term of the equation be 2.
∴ a₂ = 1, b₂ = - 3, c₂ = 2
∴ The required equation is x - 3y + 2 = 0.
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Additional Information:
1. Linear Equations in two variables:
The equation with the highest index ( degree ) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.
The general formula of linear equation in two variables is
ax + by + c = 0
Where, a, b, c are real numbers and
a ≠ 0, b ≠ 0.
2. Solution of a Linear Equation:
The value of the given variable in the given linear equation is called the solution of the linear equation.
3. Conditions and the nature of lines:
Relation between → Nature of lines
coefficients
1. a₁ / a₂ = b₁ / b₂ = c₁ / c₂ → Coincident lines
2. a₁ / a₂ ≠ b₁ / b₂ → Intersecting lines
3. a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂ → Parallel lines