Question

Check graphically whether the pair of equations 3x – 2y + 2 = 0 and 2x – y + 3 = 0, is consistent. Also, find the coordinates of the points where the graphs of the equations meet the Y-axis.
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Answer 1

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

The pair of lines are

$\rm :\longmapsto\:3x - 2y + 2 = 0$

and

$\rm :\longmapsto\:2x - y + 3 = 0$

Consider,

$\rm :\longmapsto\:3x - 2y + 2 = 0$

can be rewritten as

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

On substituting 'x = 0' in the given equation, we get

$\rm :\longmapsto\:2y = 3 \times 0 + 2$

$\rm :\longmapsto\:2y = 0 + 2$

$\rm :\longmapsto\:2y = 2$

$\bf\implies \:y = 1$

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

$\rm :\longmapsto\:2y = 3 \times 2 + 2$

$\rm :\longmapsto\:2y = 6 + 2$

$\rm :\longmapsto\:2y = 8$

$\bf\implies \:y = 4$

Substituting 'x = - 2' in the given equation, we get

$\rm :\longmapsto\:2y = 3 \times ( - 2) + 2$

$\rm :\longmapsto\:2y = - 6 + 2$

$\rm :\longmapsto\:2y = - \: 4$

$\bf\implies \:y = - \: 2$

Hᴇɴᴄᴇ,

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

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

➢ Now draw a graph using the points (0 , 1), (2 , 4) & (- 2 , - 2)

➢ See the attachment graph. [ Black line ]

Consider,

$\rm :\longmapsto\:2x - y + 3 = 0$

can be rewritten as

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

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

$\rm :\longmapsto\:y = 2 \times 0 + 3$

$\rm :\longmapsto\:y = 0 + 3$

$\bf\implies \:y = 3$

Substituting 'x = 1' in the given equation, we get

$\rm :\longmapsto\:y = 2 \times 1 + 3$

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

$\bf\implies \:y = 5$

Substituting 'x = - 1' in the given equation, we get

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

$\rm :\longmapsto\:y = - 2 + 3$

$\bf\implies \:y = 1$

Hᴇɴᴄᴇ,

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

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 3 \\ \\ \sf 1 & \sf 5 \\ \\ \sf - 1 & \sf 1 \end{array}} \\ \end{gathered}$

➢ Now draw a graph using the points (0 , 3), (1 , 5) & (-1 , 1)

➢ See the attachment graph. [ Purple line ].

From graph we concluded that,

1. As lines are intersecting, so system of equations are consistent having unique solution and solution is x = - 4 and y = - 5.

2. Line (1) meet the y - axis at (0, 1).

3. Line (2) meet the y - axis at (0, 3).