Question

Check whether the following pair of equation is consistent.
$3x+4y=2$ and $6x+8y=4$. Verify by a graphical representation.

Answer 1

If two lines consistent then lines may be intersect or coincided to each other.

If the line $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ are consistent then,

either $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$ or $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Now, the given equations are,

$3x+4y=2\Rightarrow 3x+4y-2=0$

$6x+8y=4\Rightarrow 6x+8y-4=0$

Here, $\frac{3}{6}=\frac{4}{8}=\frac{-2}{-4}=\frac{1}{2}$

Hence, the given lines are consistent. (Coincided).

Lets draw the graph,

$3x+4y=2$

Put $x=0$

$\Rightarrow 3\left(0\right)+4y=2$

$\Rightarrow 4y=2$

$\Rightarrow y=\frac{1}{2}$

$\Rightarrow (x,y)=(0,\frac{1}{2})$

Put, $y=0$

$3x+4y=2$

$\Rightarrow 3x+4\left(0\right)=2$

$\Rightarrow 3x=2$

$\Rightarrow x=\frac{2}{3}$

$\Rightarrow (x,y)=(\frac{2}{3},0)$

Therefore, the plot points of the line $3x+4y=2$ are $(0,\frac{1}{2})$ and $(\frac{2}{3},0)$.

Similarly, we get the same plot points for $6x+8y=4$ by taking the values of $x=0$ and $y=0$.

From the graph, both lines are coincided.

Hence, the given lines are consistent.