Problem 63 - Algebra and Trigonometry

Question

In Exercises 63–68, find the solution set for each system by graphing both of the system’s equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{aligned} (y-2)^{2} &=x+4 \\ y &=-\frac{1}{2} x \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations do not have any solution as none of the intersection points satisfies both equations simultaneously.

Step 1: Rewrite the Equations

The first equation \( (y-2)^{2} =x+4 \) can be rewritten as \( y = \sqrt{x + 4} + 2 \) and the second equation \( y = -\frac{1}{2} x \) remains as it is.

Step 2: Graph the Equations

Next, graph the two equations in the same coordinate system.\\For the square root function, start from (-4,2), which is the vertex. Since it is a square root function starting from 2 the graph will be quarter of a circle opening upwards.\\For the linear function with a negative slope, draw a straight line that starts from the origin (0,0) and goes downwards to the right.

Step 3: Find Points of Intersection

The solution to the system of equations are the points where the two graphs intersect. From the graph, it can be seen that the two graphs intersect at two points. They are approximately (-2,3) and (0,2).

Step 4: Verify the Solutions

Lastly, verify that these points are solutions for both equations.\\For checkpoint (-2,3), put these values into both equations: \\ For the first equation: \( (3-2)^{2} =-2+4 \) which simplifies to \( 1=2 \), which is a false statement. Therefore, (-2,3) is not a solution.\\For checkpoint (0,2), put these values into both equations:\\For the first equation: \( (2-2)^{2} =0+4 \), which simplifies to \( 0=4 \), which is again a false statement. Therefore, (0,2) is also not a solution.\\Thus, no solutions were found.

Key Concepts

Graphing

Graphing is a powerful method used to find solutions to systems of equations, especially when it's hard to solve analytically. It involves plotting equations on a graph to see where they intersect. Each equation is represented as a curve or line. By comparing their graphical representations, we can visually identify common points which, if any, are the solutions to the system.

• For non-linear equations, the graph can be a curve such as a parabola, hyperbola, or another type of curve depending on the function involved.
• For linear equations, like our second equation, the graph is a straight line.

Graphing helps us understand not only the solutions but the overall behavior of the equations involved. It eliminates a lot of guesswork since seeing the entire set visually can provide insights that equations alone might not.

Solution Set

The solution set of a system of equations consists of all the intersection points of the graphs of those equations. In essence, these are the points that satisfy all the given equations at once.

• In our exercise, we attempted to figure out where the output of each function overlaps, indicating that an x-value gives the same y-value in both equations.
• Once you identify these points from the graph, these are the candidates for the solution set, assuming the graphing is accurate and correctly identifies all intersections.

If there turns out to be no intersection, it suggests that there is no common solution to both functions. It's crucial to verify these points by substituting them back into the given equations, as demonstrated in the solution steps.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is the framework used to graph equations visually. It consists of two perpendicular axes:

• The horizontal axis is called the x-axis.
• The vertical axis is known as the y-axis.

Each point on this plane can be described by an ordered pair (x, y), where x is the value on the horizontal axis and y on the vertical axis. Using this system allows us to spatially distribute data points, thereby maintaining a visual and relational link between the different values defined by the equations.
On interacting with it, the system makes it easier to detect when and where graphs will intersect, which is essential for solving systems of equations by the graphing method.

Intersection Points

Intersection points occur when two graphs meet at certain coordinates on the xy-plane; such points provide potential solutions for a system of equations. If each point satisfies all equations involved, it belongs to the solution set. During the graphing of our exercise:

• We anticipated intersection points like (-2, 3) and (0, 2) based on our plotted graphs.
• However, on verifying these points with the original equations, neither of them held true as solutions.

Finding intersection points graphically can be observed easily but must be confirmed analytically by substituting back into the equations for validation. This ensures these points are actual solutions and not just visually misleading intersections.
This lesson underscores the importance of not only finding these points but also checking if they satisfy all involved equations.