Question

24. Draw the graph of the equations x + 3y = 6 and 2x - 3y = 12. Determine the
coordinates of the vertices of the triangle formed by these lines and the y-axis, and shade
the triangular region.
25. Draw the graphs of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the
coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade
the triangular region.



both must be drawn in same graph sheet

i will follow anyone who sends the answer plss​

Answer 1

Given:

The equations of two sets of lines:

1st set-

$x + 3y = 6\\2x-3y=12$

2nd set-

$x-y=-1\\3x+2y=12$

To find:

The vertices of the triangles formed by the above sets of lines with y-axis.

Note: A graph is attached with the solution which illustrates the graphical plot of the triangles that are shaded accordingly.

Explanation:

• To plot a line on graph, the coordinates are needed. If the coordinates are not available, they should be found.

• Here, the coordinates are not specified. Hence, the coordinates for the lines can be found by a basic method of putting "0" alternatively in x-place and in y- place.
• Consider the first set of equations:

$x+3y=6$...... (1)

$2x-3y=12$.....(2)

• In the equation (1), if we put "0" in x- place, the equation becomes:

$0+3y=6\\3y=6\\y=2$

⇒For the line equation (1), one solution is $(0,2)$.

• Now, in the same equation, if "0" is kept in y-place, then the equation becomes:

$x+3(0)=6\\x=6$

⇒For the line equation (1), the second solution is $(6,0)$.

• These solutions are the coordinates at which the line (1) meets y-axis and x-axis respectively.
• Similarly, the equation (2) also meets the x-axis and y-axis at the coordinates:

When y=0

$2x-3(0)=12\\2x=12\\x=6$

⇒For the equation (2), one solution is $(6,0)$

When x=0

$2(0)-3y=12\\-3y=12\\y=-4$

⇒For the equation (2), the other solution is $(0,-4)$.

• Using the above coordinates, the lines are drawn in the graph and a triangle is formed by these two lines with y-axis. That triangle is named one and is shaded with lines.
• The vertices of the triangle formed are $(6,0), (0,2), (0,-4)$.
• Consider the second set of equations:

$x-y=-1$......(3)

$3x+2y=12$......(4)

• For the equations (3) and (4), the similar pattern should be followed to find the coordinates.
• For the equation (3), the solutions obtained are $(0,1)$ and $(-1,0)$.
• For the equation (4), the solutions obtained are $(4,0)$ and $(0,6)$.
• Using these solution coordinates, the lines are drawn on the same graph and the triangle formed by these lines with y-axis is numbered as two and is shaded as a fill.
• The vertices of the triangle formed by the second set of lines are $(0,1), (0,6), (2,3)$.