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Chapter 5: Problem 23

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x^{2}+4 y^{2}-16=0 \\ 2 x^{2}-3 y^{2}-5=0 \end{array}\right. $$

Short Answer

Expert verified
The solutions to the system of equations are \(x = 卤\sqrt{58/13}\) and \(y = 卤\sqrt{50/52}\).

Step by step solution

01

Multiply the Equations

First, we aim to eliminate one variable by properly multiplying the equations. Here, if we multiply the first equation by 3 and the second by 2, the terms containing \(y^2\) will get cancelled when added. So, the multiplied equations are: \[9x^2 + 12y^2 - 48 = 0\] and \[4x^2 - 6y^2 - 10 = 0\]
02

Add the multiplied equations

Adding these two equations yields: \[13x^2 - 58 = 0\].
03

Solve for x

Rearrange the equation and solve for \(x\): \[x^2 = 58 / 13\] which gives \(x = 卤\sqrt{58/13}\), so \(x = 卤\sqrt{58/13}\).
04

Substitute x in either equation

Substituting \(x\) in the first equation gives: \[3(58/13) + 4y^2 - 16 = 0\]. Rearranging and simplifying for \(y\), we get: \[4y^2 = 16 - 174/13\], which further simplifies to: \[y^2 = 50/52\], so \(y = 卤\sqrt{50/52}\). Hence, \(y = 卤\sqrt{50/52}\).
05

Present the solutions

So the solutions to the system are \(x = 卤\sqrt{58/13}\), \(y = 卤\sqrt{50/52}\).

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Key Concepts

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

Systems of Equations
A system of equations is a collection of two or more equations with the same set of variables. The goal is to find values for these variables that satisfy all equations in the system simultaneously. In the given exercise, we have two equations involving quadratic terms.
  • The first equation is: \(3x^2 + 4y^2 - 16 = 0\)
  • The second equation is: \(2x^2 - 3y^2 - 5 = 0\)
Solving a system of equations can reveal points of intersection between the curves represented by these equations.
Depending on the equations, solutions might consist of ordered pairs \((x, y)\) that satisfy all equations.
Solving Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving them involves finding the values of \(x\) that make the equation true. In many systems of equations, one or more of the equations are quadratic, requiring specific solution methods.
For the exercise given, after using the addition method to eliminate \(y^2\), we have a quadratic equation:
\[13x^2 - 58 = 0\]To solve for \(x\), rearrange it as \[x^2 = \frac{58}{13}\], and take the square root to find \(x = \pm\sqrt{\frac{58}{13}}\). Often, quadratic solutions include positive and negative roots.
Variables in Algebra
Variables are symbols or letters used to represent numbers or values in equations. They are the unknowns that we aim to solve. In algebra, variables are crucial because they allow us to write general rules and relationships.
In the exercise, the variables \(x\) and \(y\) are used:
  • \(x\) and \(y\) represent unknown values we need to determine.
  • Quadratic terms, such as \(x^2\) and \(y^2\), indicate the equations are quadratic in nature.
By finding specific values of \(x\) and \(y\), we solve the equations given.
Substitution Method
The substitution method involves solving one of the equations for one variable and substituting this expression into the other equation. This process helps in reducing the system of equations to a single equation in one variable.
In our exercise, once we had \(x = \pm\sqrt{\frac{58}{13}}\), we substituted \(x\) back into one of the original equations. This allowed us to solve for the variable \(y\):
  • Substitute the value of \(x\) into the first equation to find \(y\).
  • This substitution transformed our problem into solving a simpler equation \(4y^2 = 16 - \frac{174}{13}\).
This method simplifies the process of finding solutions to variables.

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Most popular questions from this chapter

Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$ Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$

write the partial fraction decomposition of each rational expression. $$ \frac{4}{2 x^{2}-5 x-3} $$

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other nethods used in conjunction with linear programming? What are these methods? What sort of academic background, garticularly in mathematics, does this executive have? Present group report addressing these questions, emphasizing the ole of linear programming for the business.

How do you determine whether a given ordered triple is a solution of a system in three variables?

Use the two steps for solving a linear programming problem. You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?
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