Problem 45
Question
Use the graphical method to find all solutions of the system of equations, correct to two decimal places. $$\left\\{\begin{array}{l}x^{4}+16 y^{4}=32 \\\x^{2}+2 x+y=0\end{array}\right.$$
Step-by-Step Solution
Verified Answer
The system has two solutions: approximately ( 1.65, -4.72) and ( -1.65, -0.28).
1Step 1: Understand the System of Equations
We have a system of two equations: \(x^4 + 16y^4 = 32\) and \(x^2 + 2x + y = 0\). Our goal is to find the intersection points of these equations' graphs by plotting them on the coordinate plane.
2Step 2: Graph the First Equation
The equation \(x^4 + 16y^4 = 32\) represents a shape similar to an ellipse in the xy-plane. To plot it, rewrite as \(y = \pm \sqrt[4]{\frac{32 - x^4}{16}}\). Calculate several values for \(x\) and find corresponding \(y\) values to generate the graph.
3Step 3: Graph the Second Equation
The equation \(x^2 + 2x + y = 0\) can be rewritten as \(y = -x^2 - 2x\), which is a parabola opening downwards. Plot the parabola by substituting various values for \(x\) and calculating \(y\).
4Step 4: Find Points of Intersection
Identify the points where the graphs from Steps 2 and 3 intersect. These intersection points are the solutions to the system. Use two decimal places accuracy by inspecting the graph or, if needed, using numerical estimation techniques such as zooming into the graph.
5Step 5: Validate Intersection Points
For each intersection point that appears likely on the graph, substitute back into both original equations to ensure they satisfy the system. Adjust for any small inaccuracies.
Key Concepts
System of EquationsIntersection PointsCoordinate PlaneNumerical Estimation
System of Equations
A **system of equations** consists of two or more equations that share common variables. The goal is to find sets of values for these variables that satisfy all the equations simultaneously. In this exercise, the system consists of two equations:
The solution to the system is any point \((x, y)\) where the graphs of these two equations intersect. These points indicate where the values of \(x\) and \(y\) satisfy both equations at the same time.
- First equation: \(x^4 + 16y^4 = 32\)
- Second equation: \(x^2 + 2x + y = 0\)
The solution to the system is any point \((x, y)\) where the graphs of these two equations intersect. These points indicate where the values of \(x\) and \(y\) satisfy both equations at the same time.
Intersection Points
**Intersection points** occur wherever two graphs cross on the coordinate plane. In the context of solving a system of equations, intersection points represent solutions that satisfy each equation in the system.
In our exercise, these points indicate where the curve representing \(x^4 + 16y^4 = 32\) meets the parabola given by \(x^2 + 2x + y = 0\).
To find these points, we plot both equations and identify their crossing points visually. However, ensure to check these points by substituting back into the original equations to confirm they satisfy both.
In our exercise, these points indicate where the curve representing \(x^4 + 16y^4 = 32\) meets the parabola given by \(x^2 + 2x + y = 0\).
To find these points, we plot both equations and identify their crossing points visually. However, ensure to check these points by substituting back into the original equations to confirm they satisfy both.
Coordinate Plane
The **coordinate plane** is a two-dimensional plane formed by two perpendicular lines, the x-axis and the y-axis. Points on this plane are identified by (x, y) coordinates.
When graphing systems of equations, each equation translates into a curve on this plane.
For example:
When graphing systems of equations, each equation translates into a curve on this plane.
For example:
- The equation \(x^4 + 16y^4 = 32\) results in a shape similar to an ellipse, showing how different values pair together on the plane.
- \(x^2 + 2x + y = 0\) represents a downward-opening parabola, indicating another possible pairing of values.
Numerical Estimation
When finding the exact intersection points graphically, you often rely on **numerical estimation**. This involves visually inspecting where the graphs intersect and estimating the exact coordinates.
Given some graphs might intersect at non-integer values, it’s vital to estimate these points accurately, usually to two decimal places in this case.
Tools such as "zooming" in the graphical interface can improve accuracy in identifying the exact position.
Lastly, always validate by plugging these estimated points back into the original equations to ensure no errors were made during estimation.
Given some graphs might intersect at non-integer values, it’s vital to estimate these points accurately, usually to two decimal places in this case.
Tools such as "zooming" in the graphical interface can improve accuracy in identifying the exact position.
Lastly, always validate by plugging these estimated points back into the original equations to ensure no errors were made during estimation.
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