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Chapter 8: Problem 24

Solve each system. $$\begin{aligned} &x^{2}+y^{2}=5\\\ &x^{2}+4 y^{2}=14 \end{aligned}$$

Short Answer

Expert verified
Solution set is (\pm\sqrt2, \pm\sqrt3).

Step by step solution

01

- Subtract the Equations

To eliminate the term with \(x^2\), subtract the second equation from the first. \[ x^2 + y^2 - (x^2 + 4y^2) = 5 - 14 \ \Rightarrow y^2 - 4y^2 = -9 \ \Rightarrow -3y^2 = -9 \ \therefore y^2 = 3 \ \therefore y = \pm \sqrt3 . \]
02

- Substitute y back into the first equation

Using \( y = \pm \sqrt3 \), substitute \( y \) back into the first equation to find \( x \). \[ x^2 + (\pm\sqrt3)^2 = 5 \ \Rightarrow x^2 + 3 = 5 \ \Rightarrow x^2 = 2 \ \therefore x = \pm \sqrt2 . \]
03

- Write the Solution Set

Combine all the values of \( x \) and \( y \) to write the solution set: \ \((x, y) \Rightarrow (\pm \sqrt2 , \pm \sqrt3 ) . \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
When solving systems of nonlinear equations, the substitution method can be a powerful tool. This involves solving one of the equations for one variable and then substituting that expression into the other equation. This can often help simplify the problem, making it easier to solve for the variables you need. Unlike linear equations, nonlinear systems can include squares, circles, and other complex relationships, so substitution helps navigate through these complexities.

For instance, if you have equations like



    • and
    • another equation x^2 + 4y^2 = 14.
You can work with either equation to isolate and substitute. However, this method does not always simplify things right away, so it is important to carefully choose your steps.
solution set
The solution set of a system of equations is the collection of all solutions that satisfy each equation in the system. When working with nonlinear equations, solutions might not always be simple integers. Often, they can be square roots or even more complex expressions.

For the given system:
  • x^2 + y^2 = 5
  • x^2 + 4y^2 = 14
After finding y = 卤鈭�3 and substituting back in, we find the values for x and combine them:
  • When y = 鈭�3: x = 鈭�2.
  • When y = -鈭�3: x = -鈭�2.
  • When combining, we get (卤鈭�2, 卤鈭�3).
This set includes all combinations of x and y that satisfy the equations.
elimination method
The elimination method is another useful strategy to solve systems of nonlinear equations. Here, you add or subtract the equations to eliminate one of the variables. This simplifies one of the equations and enables you to solve for the other variable first. Once one variable is found, it is easier to substitute back into one of the original equations to find the second variable.

For example, given the equations:
  • x^2 + y^2 = 5
  • x^2 + 4y^2 = 14
Subtracting the second from the first:
y^2 - 4y^2 = -9
results in:
-3y^2 = -9
resulting in:
y^2 = 3
So, y = 卤鈭�3. Then substituting y back into the first equation to solve for x:
x^2 + 3 = 5
results in:
x^2 = 2
So, x = 卤鈭�2. Combining both variables gives us the full solution set as detailed earlier.

Remember to always verify the solution set with the original equations to ensure they satisfy every equation given.

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Most popular questions from this chapter

Solve each system. $$\begin{aligned} 2 x-y &=-1 \\ -2 x+z &=1 \\ y-z &=0 \end{aligned}$$

Write a system of inequalities that describes the possible solutions to each problem and graph the solution set to the system. Delicate Balance A fast food restaurant must have a minimum of 30 employees and a maximum of \(50 .\) To avoid charges of sexual bias, the company has a policy that the number of employees of one sex must never exceed the number of employees of the other sex by more than six. How many persons of each sex could be employed at this restaurant?

A system of equations can be used to find the equation of a line that goes through two points. For example, if \(y=a x+b\) goes through \((3,5),\) then a and b must satisfy \(3 a+b=5 .\) For each given pair of points, find the equation of the line \(y=a x+b\) that goes through the points by solving a system of equations. $$(-2,3),(4,-7)$$

Solve each problem by using a system of three linear equations in three variables. Cooperative Learning Write a system of three linear equations in three unknowns for which \((1 / 2,1 / 3,1 / 4)\) is the only solution. Ask a classmate to solve the system.

Solve each problem by using a system of three linear equations in three variables. Cooperative Learning Write a word problem for which a system of three linear equations in three unknowns can be used to find the solution. Ask a classmate to solve the problem.
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