Question

8. Show graphically that the following system of equations are inconsistent.

3x + 4y = 1 ;

$2x = 5 - \frac{8}{3}y$

Standard:- 10

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Answer 1

We are given a pair of linear equations in two variables, and we have to solve graphically.
We need to plot the graph of two equations. 
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The first equation is:

$\boxed{3x+4y=1} \\ \\ \implies 4y = 1-3x \\ \\ \implies y = \frac{1-3x}{4}$

Now, we need to find two points which satisfy this.

Let us put $x=-1$. We get

$y=\frac{1-3x}{4} = \frac{1-3(-1)}{4} = \frac{4}{4} = 1$

So, (-1,1) lies on the line.


We also need one other point to plot the graph of the line.

Put x=3, we get:

$y=\frac{1-3x}{4} = \frac{1-3(3)}{4} = \frac{-8}{4} = -2$

So, (3,-2) also lies on the line. 

Joining these two points, we get the graph of $3x+4y=1$

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The second equation is:

$2x = 5 - \frac{8}{3}y \\ \\ \implies 2x = \frac{15-8y}{3} \\ \\ \implies 6x=15-8y \\ \\ \implies \boxed{6x+8y=15} \\ \\ \implies 8y = 15-6x \\ \\ \implies y = \frac{15-6x}{8}$

Now, we need a couple of points on this line.

Let us put $x=-\frac{3}{2}$

$y=\frac{15-6x}{8} = \frac{15-6 \times \frac{-3}{2}}{8} = \frac{24}{8} = 3$

So, (-1.5, 3) lies on the line.


Also, let us put $y=0$

We have:

$2x=5-\frac{8}{3}y = 5 - 0 = 5 \\ \\ \implies 2x=5 \\ \\ \implies x = \frac{5}{2} = 2.5$

So, (2.5, 0) lies on the line.

We can plot the graph of $6x+8y=15$ using the two points.

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The graph is attached as an image.

Now, the graph consists of Two Parallel Lines. 

Since the lines are parallel, they are never going to intersect. And since they do not intersect, the given pair of linear equations has no solution. 
So, The equations are inconsistent.


Hence Proved.