Question

show graphically that the system of equations 2x+3y=10; 2x +5y=12 is consistent​

Answer 1

$\large\underline{\sf{Solution-}}$

Given equation of lines are

$\rm :\longmapsto\:2x + 3y = 10 - - - - (1)$

and

$\rm :\longmapsto\:2x + 5y = 12 - - - - (2)$

Consider Equation (1)

$\rm :\longmapsto\:2x + 3y = 10$

Substituting 'y = 0' in the given equation, we get

$\rm :\longmapsto\:2x + 3(0) = 10$

$\rm :\longmapsto\:2x = 10$

$\rm :\longmapsto\:x = 5$

Substituting 'y = 2' in the given equation, we get

$\rm :\longmapsto\:2x + 3(2) = 10$

$\rm :\longmapsto\:2x + 6 = 10$

$\rm :\longmapsto\:2x = 10 - 6$

$\rm :\longmapsto\:2x = 4$

$\rm :\longmapsto\:x = 2$

Substituting 'y = 4' in the given equation, we get

$\rm :\longmapsto\:2x + 3(4) = 10$

$\rm :\longmapsto\:2x + 12 = 10$

$\rm :\longmapsto\:2x = 10 - 12$

$\rm :\longmapsto\:2x = - 2$

$\rm :\longmapsto\:x = - 1$

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 5 & \sf 0 \\ \\ \sf 2 & \sf 2 \\ \\ \sf - 1 & \sf 4 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the above points

➢ See the attachment graph.

Now,

Consider equation (2)

$\rm :\longmapsto\:2x + 5y = 12$

Substituting 'y = 0' in the given equation, we get

$\rm :\longmapsto\:2x + 5(0) = 12$

$\rm :\longmapsto\:2x = 12$

$\rm :\longmapsto\:x = 6$

Substituting 'y = 2' in the given equation, we get

$\rm :\longmapsto\:2x + 5(2) = 12$

$\rm :\longmapsto\:2x + 10= 12$

$\rm :\longmapsto\:2x= 12 - 10$

$\rm :\longmapsto\:2x= 2$

$\rm :\longmapsto\:x= 1$

Substituting 'y = 4' in the given equation, we get

$\rm :\longmapsto\:2x + 5(4) = 12$

$\rm :\longmapsto\:2x + 20= 12$

$\rm :\longmapsto\:2x= 12 - 20$

$\rm :\longmapsto\:2x= - 8$

$\rm :\longmapsto\:x= - 4$

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 6 & \sf 0 \\ \\ \sf 1 & \sf 2 \\ \\ \sf - 4 & \sf 4 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using these points.

➢ See the attachment graph.

From graph,

We conclude that the given lines are intersecting,

So, system of equation is consistent.