Problem 60
Question
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. $$ \left\\{\begin{array}{r} {4 x^{2}+y^{2}=4} \\ {y^{2}-4 x^{2}=4} \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The intersections of the graphs of the two given equations provide the solution set.
1Step 1: Graphing the First Equation
First, arrange the first equation \(4 x^{2}+y^{2}=4\) into the standard form of ellipse equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). It becomes \(x^{2} + \frac{y^{2}}{4} = 1\). This means that this ellipse has a radius 1 along x axis and 2 along y axis. Now sketch this ellipse onto a graph paper.
2Step 2: Graphing the Second Equation
Arrange the second equation \(y^{2}-4 x^{2}=4\) into the standard form of hyperbola equation \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). It becomes \(\frac{y^{2}}{4} - x^{2} = 1\). This means this hyperbola opens upwards and downwards with a vertical transverse axis of length 4 and conjugate axis of length 1. Now, sketch this hyperbola on the same graph paper.
3Step 3: Identifying Intersection Points
Locate the points where the ellipse and the hyperbola intersect. These points are the solution set to the system of equations.
4Step 4: Checking the Solutions
Finally, to validate these solutions, substitute the values of x and y from the points of intersection back into both of the original equations. If both equations hold true, then the solutions are correct.
Key Concepts
Graphing Quadratic EquationsEllipse and Hyperbola IntersectionSystem of Quadratic Equations
Graphing Quadratic Equations
Graphing quadratic equations is essential for visualizing the solutions to quadratic equations and systems involving them. This process involves plotting a curve, typically a parabola, ellipse, or hyperbola, on a coordinate plane.
Most quadratic equations can be written in the form \(ax^2+bx+c=0\), depicting a parabola when graphed. However, when equations involve both \(x^2\) and \(y^2\), they represent ellipses or hyperbolas, depending on the signs of the terms. To graph these curves, we rewrite them in a standard form where the relationship between \(x\) and \(y\) is clearly seen, and their respective shapes and orientations are understood.
For example, in the first step of the provided solution, the equation \(4x^2+y^2=4\) is rearranged to resemble the standard form of an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). This adjustment allows us to identify the lengths of the major and minor axes and the coordinates of the center of the ellipse, making the graphing process more straightforward. Similarly, understanding that a hyperbola equation has a minus sign between the \(x^2\) and \(y^2\) terms is key to graphing it accurately.
Most quadratic equations can be written in the form \(ax^2+bx+c=0\), depicting a parabola when graphed. However, when equations involve both \(x^2\) and \(y^2\), they represent ellipses or hyperbolas, depending on the signs of the terms. To graph these curves, we rewrite them in a standard form where the relationship between \(x\) and \(y\) is clearly seen, and their respective shapes and orientations are understood.
For example, in the first step of the provided solution, the equation \(4x^2+y^2=4\) is rearranged to resemble the standard form of an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). This adjustment allows us to identify the lengths of the major and minor axes and the coordinates of the center of the ellipse, making the graphing process more straightforward. Similarly, understanding that a hyperbola equation has a minus sign between the \(x^2\) and \(y^2\) terms is key to graphing it accurately.
Ellipse and Hyperbola Intersection
The intersection of an ellipse and a hyperbola in a system of equations is depicted graphically as the points where their respective curves cross each other on the coordinate plane. To find these points, both shapes need to be graphed accurately according to their standard forms.
An ellipse has the standard form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a\) and \(b\) define the lengths of the axes. Meanwhile, a hyperbola has the standard form \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\), with \(a\) as the distance from the center to the vertices, and \(b\) indicating the distance to the asymptotes.
To locate the intersection points, first graph the given ellipse and hyperbola on the same coordinate system. Pay attention to the lengths and orientations of the axes. When the curves are overlaid, their intersections are observed. Those points are the solutions to the system of quadratic equations, provided they satisfy both the original equations, as seen in the step-by-step solution's verification.
An ellipse has the standard form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a\) and \(b\) define the lengths of the axes. Meanwhile, a hyperbola has the standard form \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\), with \(a\) as the distance from the center to the vertices, and \(b\) indicating the distance to the asymptotes.
To locate the intersection points, first graph the given ellipse and hyperbola on the same coordinate system. Pay attention to the lengths and orientations of the axes. When the curves are overlaid, their intersections are observed. Those points are the solutions to the system of quadratic equations, provided they satisfy both the original equations, as seen in the step-by-step solution's verification.
System of Quadratic Equations
A system of quadratic equations consists of two or more equations that involve quadratic expressions. Solving such systems can provide multiple sets of solutions, as quadratic equations are capable of having more than one root.
When solving systems of quadratic equations graphically, we visually interpret the points of intersection between the graphs of each equation. In the example problem, the system consists of an ellipse and a hyperbola. Graphing these curves reveals their points of intersection, which are the solutions to the system.
It's of utmost importance to check the solutions by plugging the intersection points back into the original equations. If the coordinates satisfy all the equations in the system, they are considered the correct solutions. The graphical method is very useful because it not only provides a visual representation of the solutions but also confirms the existence of solutions in real-number format, which can be particularly useful for complex shapes like ellipses and hyperbolas.
When solving systems of quadratic equations graphically, we visually interpret the points of intersection between the graphs of each equation. In the example problem, the system consists of an ellipse and a hyperbola. Graphing these curves reveals their points of intersection, which are the solutions to the system.
It's of utmost importance to check the solutions by plugging the intersection points back into the original equations. If the coordinates satisfy all the equations in the system, they are considered the correct solutions. The graphical method is very useful because it not only provides a visual representation of the solutions but also confirms the existence of solutions in real-number format, which can be particularly useful for complex shapes like ellipses and hyperbolas.
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