Problem 20 - Algebra 2

Question

Find the coordinates of the vertices of the figure formed by each system of inequalities. $$ \begin{array}{l}{y \geq 0} \\ {x \geq 0} \\ {x+2 y \leq 8}\end{array} $$

Step-by-Step Solution

Answer

The vertices of the region are (0, 0), (8, 0), and (0, 4).

Step 1: Identify the Inequalities

We need to solve the system of inequalities: \( y \geq 0 \), \( x \geq 0 \), and \( x + 2y \leq 8 \). These inequalities will form a region in the first quadrant of the coordinate plane.

Step 2: Graph the Boundary Line of \(x + 2y = 8\)

Start by finding where the line \( x + 2y = 8 \) intersects the axes. Set \( y = 0 \) and solve for \( x \) to find the x-intercept, which gives \( x = 8 \). Then, set \( x = 0 \) and solve for \( y \) to find the y-intercept, yielding \( y = 4 \). This line goes through the points (8, 0) and (0, 4).

Step 3: Determine the Feasible Region

The inequality \( x + 2y \leq 8 \) means that the region must be below and on the line \( x + 2y = 8 \). The inequalities \( y \geq 0 \) and \( x \geq 0 \) restrict the region to the first quadrant.

Step 4: Identify the Vertices of the Feasible Region

The vertices of the feasible region are the points of intersection of the lines and the axes. These are (0, 0), where \( y = 0 \) and \( x = 0 \); (8, 0), where \( x + 2y = 8 \) and \( y = 0 \); and (0, 4), where \( x + 2y = 8 \) and \( x = 0 \). Thus, the vertices are (0, 0), (8, 0), and (0, 4).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that helps us understand geometric concepts using a coordinate system. In this system, we represent points, lines, and curves on a plane using pairs of numbers called coordinates. This is typically done on a two-dimensional plane known as the Cartesian plane.

In the Cartesian plane, each point is identified by an ordered pair \(x, y\), where \(x\) and \(y\) are the coordinates that specify the point's position. The x-coordinate tells us how far along the horizontal axis the point is, while the y-coordinate indicates the point's position along the vertical axis.

When working with systems of inequalities, coordinate geometry allows us to visually represent the solutions by shading areas that satisfy all given inequalities. This visual representation can greatly aid our understanding of how different sets of conditions interact.

Feasible Region

A feasible region is the area or set of points that satisfy all the given inequalities in a system. It is the 'solution space' where all conditions are true. In the context of coordinate geometry, this region is typically bounded by lines, which are the graphical representation of linear inequalities.

For the exercise provided, the feasible region is determined by:

• \( y \geq 0 \): The area above the x-axis.
• \( x \geq 0 \): The area to the right of the y-axis.
• \( x + 2y \leq 8 \): The area below or on the line where this equation is true.

When these individual regions are imposed on each other, the overlapping area is the feasible region where all inequalities hold simultaneously.

This region often creates a shape, such as a triangle or polygon, depending on the equations involved. It's important in optimization problems to identify this region since any solution must lie within this boundary.

Vertex

The term 'vertex' generally refers to a corner point in a geometric shape. In terms of a feasible region formed by linear inequalities, vertices are the points where the boundary lines intersect.

In this exercise, the vertices are significant because they denote the corners of the feasible region. These vertices occur at the intersections of the lines \(x + 2y = 8\), \(y = 0\), and \(x = 0\). By calculating these intersections, we find the vertices of the region, which are:

• \((0, 0)\): Intersection of \(x = 0\) and \(y = 0\)
• \((8, 0)\): Intersection of \(x + 2y = 8\) and \(y = 0\)
• \((0, 4)\): Intersection of \(x + 2y = 8\) and \(x = 0\)

Finding these vertices is crucial in many applications, such as linear programming, where optimal values occur at these points.

Linear Inequalities

Linear inequalities are mathematical expressions that relate linear functions using inequality symbols, such as \(<\), \(>\), \(\leq\), or \(\geq\). These inequalities describe areas within a coordinate plane where certain conditions are true.

Unlike linear equations, which describe straight lines, linear inequalities describe regions on one side of a line. For example, the inequality \(x + 2y \leq 8\) creates a region that includes all points (x, y) that satisfy the inequality on and below the line \(x + 2y = 8\).

Working with linear inequalities involves:

• Graphing the boundary lines by converting inequalities to equalities.
• Determining which side of the line satisfies the inequality by testing points.
• Shading the region that fulfils the inequality conditions.

This approach helps in defining the feasible region, as seen in systems of inequalities, where the solutions form an area bounded by these linear inequalities.