X+y less than or equal to 2 and x-y that is less than or equal to 1 . Find the coordinates of the vertices.

Question

alenahelenash · Accepted Answer

To find the coordinates of the vertices of the region defined by the inequalities

x + y \leq 2

and

x - y \leq 1

, we can start by graphing the boundary lines of the region.
For the inequality

x + y \leq 2

, we draw the line

x + y = 2

. To find the coordinates of the points where this line intersects the axes, we set

x

and

y

to 0 separately and solve for the other variable.
When

x = 0

, we have

0 + y = 2

, which gives

y = 2

. So one point on the line is

(0, 2)

.
When

y = 0

, we have

x + 0 = 2

, which gives

x = 2

. So another point on the line is

(2, 0)

.
Joining these two points, we can draw the line

x + y = 2

.
Next, for the inequality

x - y \leq 1

, we draw the line

x - y = 1

. Similarly, we find the coordinates of the points where this line intersects the axes.
When

x = 0

, we have

0 - y = 1

, which gives

y = - 1

. So one point on the line is

(0, - 1)

.
When

y = 0

, we have

x - 0 = 1

, which gives

x = 1

. So another point on the line is

(1, 0)

.
Joining these two points, we can draw the line

x - y = 1

.
The region defined by the inequalities is the area below the line

x + y = 2

and below the line

x - y = 1

. We are interested in finding the vertices of this region.
To find the vertices, we look for the points where the lines intersect. The vertices of the region are the intersection points of these lines.
Solving the system of equations formed by the two lines:

{\begin{matrix} x + y = 2 \\ x - y = 1 \end{matrix}

We can add the equations together to eliminate

y

:

(x + y) + (x - y) = 2 + 1

This simplifies to:

2 x = 3

Solving for

x

:

x = \frac{3}{2}

Substituting this value of

x

back into one of the equations, let's use

x + y = 2

, we can find

y

:

\frac{3}{2} + y = 2

y = \frac{1}{2}

Therefore, the coordinates of the intersection point (vertex) are

(\frac{3}{2}, \frac{1}{2})

.
Hence, the coordinates of the vertices of the region defined by the inequalities

x + y \leq 2

and

x - y \leq 1

are

(0, - 1)

,

(\frac{3}{2}, \frac{1}{2})

, and

(2, 0)

.

X+y less than or equal to 2 and

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