For which value of K will the equations x + 2y = 7 and 2x + ky = 14 coincide:

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  1. 2
  2. 3
  3. 4
  4. None of these 

Answer (Detailed Solution Below)

Option 3 : 4
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Detailed Solution

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Concept Used:

Equation of a Line: y = mx + b (slope-intercept form)

Where m is the slope and b is the y - intercept. Also, for any two equation  Ax + By = C and Dx + Ey = F

to have the graph as coincide lines, the ratio of the coefficient of each variables as well as constant needs to be equal i.e.,

A/D = B/E = C/F

Explanation:

We have two equations x + 2y = 7 and 2x + ky = 14 which coincide. 

First, let's rewrite the second equation in the form of y = mx + b (slope-intercept form), where m is the slope and b is the y-intercept:

2x + ky = 14

Subtract 2x from both sides:

ky = -2x + 14

Now, divide both sides by k:

y = (-2/k)x + (14/k)

Now, we can compare the two equations:

x + 2y = 7
y = (-2/k)x + (14/k)

For these two equations to coincide, their slopes and y-intercepts must be the same.

The slope of the first equation is -1/2.
The slope of the second equation is -2/k.

So, to make the slopes equal, we set:

-1/2 = -2/k ⇒ 2/k = 1/2

k/2 = 2 ⇒  k = 4

So, for the two equations to coincide, k must be equal to 4.

Alternate: 

For any two equation

Ax + By = C
Dx + Ey = F

to have the graph as coincide lines, the ratio of the coefficient of each variables as well as constant needs to be equal i.e.,

A/D = B/E = C/F

Thus for equation x + 2y = 7 and 2x + ky = 14 to coincide,

1/2 = 2/k = 7/14

⇒ 1/2 = 2/k

⇒ k = 4

Thus, the required value of k is 4.

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