Solved on Nov 07, 2023

Find the coordinates of vertex A of the triangle formed by the inequalities: $x \geq 3, y \geq 2, x + y \leq 8$ .

STEP 1

Assumptions1. The given system of inequalities represents a polygon in the form of a triangle. . Each side of the triangle corresponds to the inequality with the same number.
3. The vertex $\mathbf{A}$ is located at the90 degree angle of the triangle.

STEP 2

The vertex $\mathbf{A}$ is the intersection point of the lines represented by inequalities (1) and (2). The lines are represented by the equations\begin{align} (1) & \quad x = \\ (2) & \quad y =2 \\ \end{align}

STEP 3

Since the lines are parallel to the axes, the intersection point $\mathbf{A}$ has the coordinates $(3,2)$ .

STEP 4

However, we need to ensure that this point also satisfies the third inequality. Substitute $(3,2)$ into inequality (3):
$x + y \leq8$

STEP 5

Substitute the values of $x$ and $y$ into the inequality $3 +2 \leq8$

STEP 6

implify the left-hand side $5 \leq8$

STEP 7

Since $5 \leq$ is true, the point $(3,2)$ satisfies all the inequalities, so the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .
Hence, the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .

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Find the coordinates of vertex A of the triangle formed by the inequalities: $x \geq 3, y \geq 2, x + y \leq 8$ .

STEP 1

Assumptions1. The given system of inequalities represents a polygon in the form of a triangle. . Each side of the triangle corresponds to the inequality with the same number.
3. The vertex $\mathbf{A}$ is located at the90 degree angle of the triangle.

STEP 2

The vertex $\mathbf{A}$ is the intersection point of the lines represented by inequalities (1) and (2). The lines are represented by the equations\begin{align} (1) & \quad x = \\ (2) & \quad y =2 \\ \end{align}

STEP 3

Since the lines are parallel to the axes, the intersection point $\mathbf{A}$ has the coordinates $(3,2)$ .

STEP 4

However, we need to ensure that this point also satisfies the third inequality. Substitute $(3,2)$ into inequality (3):
$x + y \leq8$

STEP 5

Substitute the values of $x$ and $y$ into the inequality $3 +2 \leq8$

STEP 6

implify the left-hand side $5 \leq8$

STEP 7

Since $5 \leq$ is true, the point $(3,2)$ satisfies all the inequalities, so the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .
Hence, the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .

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Find the coordinates of vertex A of the triangle formed by the inequalities: x≥3,y≥2,x+y≤8x \geq 3, y \geq 2, x + y \leq 8x≥3,y≥2,x+y≤8.

STEP 1

Assumptions1. The given system of inequalities represents a polygon in the form of a triangle. . Each side of the triangle corresponds to the inequality with the same number.3. The vertex A\mathbf{A}A is located at the90 degree angle of the triangle.

STEP 2

The vertex A\mathbf{A}A is the intersection point of the lines represented by inequalities (1) and (2). The lines are represented by the equations\begin{align*} (1) & \quad x = \\ (2) & \quad y =2 \\ \end{align*}

STEP 3

Since the lines are parallel to the axes, the intersection point A\mathbf{A}A has the coordinates (3,2)(3,2)(3,2).

STEP 4

However, we need to ensure that this point also satisfies the third inequality. Substitute (3,2)(3,2)(3,2) into inequality (3):x+y≤8x + y \leq8x+y≤8

STEP 5

Substitute the values of xxx and yyy into the inequality3+2≤83 +2 \leq83+2≤8

STEP 6

implify the left-hand side5≤85 \leq85≤8

STEP 7

Since 5≤5 \leq5≤ is true, the point (3,2)(3,2)(3,2) satisfies all the inequalities, so the coordinates of vertex A\mathbf{A}A are (3,2)(3,2)(3,2).Hence, the coordinates of vertex A\mathbf{A}A are (3,2)(3,2)(3,2).

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Find the coordinates of vertex A of the triangle formed by the inequalities: $x \geq 3, y \geq 2, x + y \leq 8$ .

Assumptions1. The given system of inequalities represents a polygon in the form of a triangle. . Each side of the triangle corresponds to the inequality with the same number.
3. The vertex $\mathbf{A}$ is located at the90 degree angle of the triangle.

The vertex $\mathbf{A}$ is the intersection point of the lines represented by inequalities (1) and (2). The lines are represented by the equations\begin{align} (1) & \quad x = \\ (2) & \quad y =2 \\ \end{align}

Since the lines are parallel to the axes, the intersection point $\mathbf{A}$ has the coordinates $(3,2)$ .

However, we need to ensure that this point also satisfies the third inequality. Substitute $(3,2)$ into inequality (3):
$x + y \leq8$

Substitute the values of $x$ and $y$ into the inequality $3 +2 \leq8$

implify the left-hand side $5 \leq8$

Since $5 \leq$ is true, the point $(3,2)$ satisfies all the inequalities, so the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .
Hence, the coordinates of vertex $\mathbf{A}$ are $(3,2)$ .