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Pair of Linear Equations in Two Variables

Short Answer Type

161. For what value of k, the pair of linear equations 3x - ky + 7 = 0,x - 2y + 5 = 0 has unique solution.

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162. Given below are three equations. Two of them have infinite solutions and two of them have no solution. State the two pairs :

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163. Show that x = 1 and y = 1 is a solution of the pair of equations 2x + 3y = 5. 5x - 2y = 3.

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164.

Write whether the following pair of linear equations is consistent or not.

x + y = 14
x - y = 4

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165.

Find the value of k so that the following system of equations has infinite solutions:

3x - y - 5 = 0; 6x - 2y + k = 0

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166. Find the value of k for which the pair of equations kx - 4y = 3; 6x - 12y = 9 has an infinite number of solution.

We have,
kx - 4y = 3; 6x - 12y = 9
Here, a₁ = k, b = -4,c₁ = 3
a² = 6, b₂ = - 12, c₂ = 9
For infinitely many solutions,
$straight a subscript 1 over straight a subscript 2 equals straight b subscript 1 over straight b subscript 2 equals straight c subscript 1 over straight c subscript 2 rightwards double arrow space space space space space space space space space space space space space space straight k over 6 equals fraction numerator negative 4 over denominator negative 12 end fraction equals 3 over 9 rightwards double arrow space space space space space space space space space space space space space space space straight k over 6 equals 1 third equals 1 third rightwards double arrow space space space space space space space space space space space space space space space space straight k over 6 equals 1 third rightwards double arrow 3 straight k space rightwards double arrow box enclose straight k equals 2 end enclose space space space$
Hence, the given system of equation has infinitely many solution if, k = 2.

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167. Check whether x = 3, y = -1 and x = -6, y = 5 are solutions of the system of linear equations 2x + 3y = 3 and 6x + 9y - 9 = 0. Does the given system has an infinite number of solutions ? Give reason.

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168. Obtain the condition for the following system of linear equations to have a unique solution ax + by = c and lx + my = n.

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169.

Given the linear equation 2a + 3y - 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is :

(i) parallel lines

(ii) intersecting lines

(iii) coincident lines.

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170. For what value of k, the following pair of linear equations has infinitely many solutions?

10x + 5y - (k - 5) = 0
20x + 10y - k = 0.

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