Chapter 6.4, Problem 26PPS

a.

To determine

To determine the co-ordinates of the vertex

Answer to Problem 26PPS

The co-ordinate of the vertex is (4, 1).

Explanation of Solution

Given:

The vertex of the triangle is point the intersection of the two given lines.

Equations:

  $x+2y=6$

  $2x+y=9$

Calculation:

Let,

  $\begin{array}{l}x+2y=\mathrm{6.................}(i)\\ 2x+y=\mathrm{9...................}(ii)\end{array}$

Now, Subtract equation ( ii ) from $2\times$ equation ( i ),

  $\begin{array}{l}\Rightarrow 2(x+2y)-(2x+y)=2\times 6-9\\ \Rightarrow 2x+4y-2x-y=12-9\\ \Rightarrow 3y=3\\ \Rightarrow y=1\end{array}$

Substituting the value of y in equation ( i ),

  $\begin{array}{l}\Rightarrow x+2\times 1=6\\ \Rightarrow x=4\end{array}$

Conclusion:

Hence, the co-ordinate of the vertex is (4, 1).

b.

To determine

To draw the graph of the given lines and identify the vertex of the triangle

Explanation of Solution

Given:

  $x+2y=6$

  $2x+y=9$

Calculation of graph:

Consider, $x+2y=6$

Values of x	Values of y
0	3
2	2
4	1
-2	4
-4	5

Using the above table, the graph can be plotted.

Graph:

Calculation of graph:

Consider $2x+y=9$

Values of x	Values of y
0	9
3	3
2	5
1	7
-1	11

Using the above table, the graph can be plotted.

Graph:

Plotting both the graphs on the same co-ordinate plane:

Interpretation:

From the above graph, it is clear that, the point of intersection of lines is (4, 1), which is the solution to the given equations.

Conclusion:

Hence, the vertex of the triangle is (4, 1).

c.

To determine

To draw $x-y=-3$ on the previous graph

Explanation of Solution

Given:

  $x+2y=6$

  $2x+y=9$

  $x-y=-3$

Calculation of graph:

Consider, $x+2y=6$

Values of x	Values of y
0	3
2	2
4	1
-2	4
-4	5

Using the above table the graph can be plotted.

Graph:

Calculation of graph:

Consider, $2x+y=9$

Values of x	Values of y
0	9
3	3
2	5
1	7
-1	11

Using the above table the graph can be plotted.

Graph:

Calculation of graph:

Consider, $x-y=-3$

Values of x	Values of y
0	3
1	2
2	1
3	6
4	7

Using the above table the graph can be plotted.

Graph:

Plotting all the graphs on the same co-ordinate plane:

Interpretation:

A triangle can be seen in the graph with vertices: (0, 3), (2, 5) and (4, 1).

Conclusion:

Hence, a triangle is formed with the intersections points of the given lines.

d.

To determine

To name the other two vertices

Answer to Problem 26PPS

(2, 5) and (0, 3)

Explanation of Solution

Consider the following graph from sub-part(c).

A triangle can be seen in the graph with vertices: (0, 3), (2, 5) and (4, 1).

So, the other two vertices are: (0, 3) and (2, 5).