Problem 16
Question
Solve each system. $$\begin{array}{c} 5 x^{2}-y^{2}=16 \\ x^{2}+y^{2}=14 \end{array}$$
Step-by-Step Solution
Verified Answer
The solutions of the given system are \((\sqrt{5}, 3)\), \((\sqrt{5}, -3)\), \((- \sqrt{5}, 3)\), and \((- \sqrt{5}, -3)\).
1Step 1: Express one variable in terms of another in the second equation
We express \(y^2\) in terms of \(x^2\) from the second equation:
\(y^2 = 14 - x^2\)
2Step 2: Substitute the expression to the first equation
We substitute the expression for \(y^2\) found in the previous step into the first equation:
\(5x^2 - (14 - x^2) = 16\)
3Step 3: Solve for x
Now we solve the equation for x:
\(5x^2 - 14 + x^2 = 16\)
\(6x^2 = 30\)
\(x^2 = 5\)
\(x = \pm\sqrt{5}\)
4Step 4: Solve for y
Next, we plug the values of x into the expression for \(y^2\) to find the corresponding values of y:
1. For \(x = \sqrt{5}\):
\(y^2 = 14 - (\sqrt{5})^2 = 14 - 5\)
\(y^2 = 9\)
\(y = \pm3\)
2. For \(x = -\sqrt{5}\):
\(y^2 = 14 - (-\sqrt{5})^2 = 14 - 5\)
\(y^2 = 9\)
\(y = \pm3\)
5Step 5: Determine the solutions
From Steps 3 and 4, we found the corresponding values of x and y which satisfy both given equations. The solutions are:
1. \(x = \sqrt{5}, y = 3\)
2. \(x = \sqrt{5}, y = -3\)
3. \(x = -\sqrt{5}, y = 3\)
4. \(x = -\sqrt{5}, y = -3\)
Consequently, the solutions of the given system are \((\sqrt{5}, 3)\), \((\sqrt{5}, -3)\), \((- \sqrt{5}, 3)\), and \((- \sqrt{5}, -3)\).
Key Concepts
Substitution MethodQuadratic EquationsSystem of Equations
Substitution Method
The substitution method is a powerful tool for solving systems of equations, especially when working with linear or quadratic equations. It involves solving one equation for one variable, and then substituting that expression into the other equation. This method is particularly useful for eliminating variables and simplifying the solving process.
In our problem, we start by rearranging the second equation to find an expression for one of the variables. This makes it easier to substitute and solve the quadratic system. In this case, we express \(y^2\) in terms of \(x^2\), resulting in \(y^2 = 14 - x^2\).
Once you have the expression, you substitute it into the first equation, effectively reducing the number of variables. After substituting, the equation becomes easier to solve since it now involves only one variable instead of two. This allows for a quicker resolution of the given system.
In our problem, we start by rearranging the second equation to find an expression for one of the variables. This makes it easier to substitute and solve the quadratic system. In this case, we express \(y^2\) in terms of \(x^2\), resulting in \(y^2 = 14 - x^2\).
Once you have the expression, you substitute it into the first equation, effectively reducing the number of variables. After substituting, the equation becomes easier to solve since it now involves only one variable instead of two. This allows for a quicker resolution of the given system.
Quadratic Equations
Quadratic equations are equations that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Solving quadratic equations involves finding the values of \(x\) that satisfy the equation.
In our exercise, the system involves two quadratic equations: \(5x^2 - y^2 = 16\) and \(x^2 + y^2 = 14\). When solving quadratic systems, you might find more than one solution for the variables involved. For instance, in this case, the solution yields the values \(x = \pm \sqrt{5}\) and \(y = \pm 3\).
These solutions are found using common methods like substitution, factoring, or using the quadratic formula. Understanding how to carefully manipulate quadratic equations while respecting algebraic rules is crucial when dealing with these types of mathematical challenges.
In our exercise, the system involves two quadratic equations: \(5x^2 - y^2 = 16\) and \(x^2 + y^2 = 14\). When solving quadratic systems, you might find more than one solution for the variables involved. For instance, in this case, the solution yields the values \(x = \pm \sqrt{5}\) and \(y = \pm 3\).
These solutions are found using common methods like substitution, factoring, or using the quadratic formula. Understanding how to carefully manipulate quadratic equations while respecting algebraic rules is crucial when dealing with these types of mathematical challenges.
System of Equations
A system of equations consists of two or more equations with a common set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously.
For this specific task, the system is composed of two quadratic equations, which means it requires a careful approach like substitution or elimination to solve. When facing a system where both equations are quadratic, you can encounter multiple solutions because each quadratic equation can have up to two real solutions.
By expressing one equation in terms of a single variable using methods like substitution, you simplify the system, which helps reach the solution faster. In our problem, substituting the expression derived for \(y^2\) into the first equation smoothly transforms it into one-manageable equation with variable \(x\), eventually revealing the possible values for \(x\) and \(y\). This was essential in determining the solution set \( (\sqrt{5}, 3), (\sqrt{5}, -3), (-\sqrt{5}, 3), \) and \((-\sqrt{5}, -3)\).
For this specific task, the system is composed of two quadratic equations, which means it requires a careful approach like substitution or elimination to solve. When facing a system where both equations are quadratic, you can encounter multiple solutions because each quadratic equation can have up to two real solutions.
By expressing one equation in terms of a single variable using methods like substitution, you simplify the system, which helps reach the solution faster. In our problem, substituting the expression derived for \(y^2\) into the first equation smoothly transforms it into one-manageable equation with variable \(x\), eventually revealing the possible values for \(x\) and \(y\). This was essential in determining the solution set \( (\sqrt{5}, 3), (\sqrt{5}, -3), (-\sqrt{5}, 3), \) and \((-\sqrt{5}, -3)\).
Other exercises in this chapter
Problem 16
Solve each system. $$\begin{aligned} 5 x^{2}-y^{2} &=16 \\ x^{2}+y^{2} &=14 \end{aligned}$$
View solution Problem 16
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The Winter Olympics have been held a total of 21 times on the continents of North America, Europe, and Asia. The number of European sites is 5 more than the tot
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Solve the exponential equation algebraically. Then check using a graphing calculator. $$e^{t}=1000$$
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