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Pair of Linear Equations in Two variables — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsPair of Linear Equations in Two variables5 Marks
Question
Is the pair of linear equation consistent/inconsistent? If consistent, obtain the solution graphically: $2x + y - 6 = 0; 4x - 2y - 4 = 0$
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Answer

$2x + y - 6 = 0 ...(1)$
$4x - 2y - 4 = 0 ...(2)$
Here, $a_1 = 2, b_1 = 1, c_1 = -6$
$a_2 = 4, b_2 = -2, c_2 = -4$
We see that $\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}}$
Hence, the lines represented by the equations (1) and (2) are intersecting.
Therefore equation (1) and (2) have exactly one (unique) solution i.e., the given pair of linear equation is consistent. Graphical representation. We draw the graphs of the equations (1) and (2) by finding two solutions for each of the equations.
These two solution of the equations (1) and (2) given below in table 1 and 2 respectively.
For equation (1)
$2x + y - 6 = 0$
$\Rightarrow$ $y = -2x + 6$
Table 1 of solutions
x 0 3
y 6 0
For equation (2)
4x - 2y - 4 = 0
$\Rightarrow$ 2y = 4x - 4
$\Rightarrow y = \frac{{4x - 4}}{2} \Rightarrow$ y = 2x - 2
Table 2 of solutions
x 0 1
y -2 0
We plots the points A(0, 6) and B(3, 0) on a graph paper and join these points to form the line AB representing the equation (1) as shown in the figure.
Also. we plot the points C(0, -2) and D(1, 0) on the same graph paper and join these points to form the line CD representing the equation (2) as shown in the same figure. In the figure, we observe that the same lines intersect
at the point P(2, 1). So x = 2 and y = 1 is the required unique solution of the pair of linear equations formed.
Verification : substituting x = 2 and y = 1 in (1) and (2) we find that both the equations are satisfied as shown below:
2x + y - 6 = 2(2) + 2 = 6
4x - 2y - 4 = 4(2) - 2(2) - 4 = 0
This verifies the solution.

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