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Chapter 7: Problem 11

For the following exercises, solve the system of nonlinear equations using elimination. $$ \begin{array}{l} 4 x^{2}-9 y^{2}=36 \\ 4 x^{2}+9 y^{2}=36 \end{array} $$

Short Answer

Expert verified
The solutions are \((3, 0)\) and \((-3, 0)\).

Step by step solution

01

Understanding the Problem

We have a system of two nonlinear equations:\[ 4x^2 - 9y^2 = 36 \]\[ 4x^2 + 9y^2 = 36 \]The goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously.
02

Adding the Equations

Add the two equations to eliminate \(y^2\):\[(4x^2 - 9y^2) + (4x^2 + 9y^2) = 36 + 36\]This simplifies to:\[8x^2 = 72\]
03

Solving for x

Solve \(8x^2 = 72\) for \(x^2\):\[8x^2 = 72 \x^2 = \frac{72}{8} = 9\]Take the square root of both sides:\[x = \pm 3\]
04

Subtracting the Equations

Subtract the second equation from the first to eliminate \(x^2\):\[(4x^2 - 9y^2) - (4x^2 + 9y^2) = 36 - 36\]This simplifies to:\[-18y^2 = 0\]
05

Solving for y

Solve \(-18y^2 = 0\) for \(y^2\):\[y^2 = 0\]Take the square root:\[y = 0\]
06

Verifying the Solutions

Substitute \(x = 3\) and \(x = -3\) into the original equations to verify:For \(x = 3\):1. \(4(3)^2 - 9(0)^2 = 36\)2. \(4(3)^2 + 9(0)^2 = 36\)Both equations are satisfied.For \(x = -3\):1. \(4(-3)^2 - 9(0)^2 = 36\)2. \(4(-3)^2 + 9(0)^2 = 36\)Both equations are satisfied.

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Key Concepts

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

Elimination Method
The elimination method is a popular technique used to solve systems of equations. This method involves combining equations to eliminate one of the variables. Once a variable is eliminated, the remaining equation, which now involves only one variable, can be solved more easily.
In the given problem:
  • We start with two nonlinear equations: \(4x^2 - 9y^2 = 36\) and \(4x^2 + 9y^2 = 36\).
  • Adding these equations eliminates \(y^2\), leading to a simpler equation: \(8x^2 = 72\).
  • Subtracting the second equation from the first eliminates \(x^2\), resulting in \(-18y^2 = 0\).
    • This strategic approach efficiently provides isolated equations for each variable, allowing us to solve them independently.
System of Equations
A system of equations is a set of two or more equations with the same variables. Solving a system means finding a set of variable values that satisfies all equations in the system simultaneously.
There are various methods for solving systems, such as:
  • Graphing
  • Substitution
  • Elimination
For nonlinear systems like the one we have, the elimination method can be particularly effective because it simplifies the problem by reducing the number of variables in each equation. By adding and subtracting equations:
  • We obtained simpler quadratic equations focused on one variable at a time.
  • This led to finding simple solutions for \(x\) and \(y\) which satisfy both original equations.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree, generally expressed as \(ax^2 + bx + c = 0\). In the system we are solving, both equations are quadratic in terms of \(x\) and \(y\).
Here's how we approached these quadratic equations:
  • After applying the elimination method, we simplified the quadratic into \(8x^2 = 72\), which can be solved by first dividing through by 8.
  • The remaining equation \(x^2 = 9\) gives two potential solutions for \(x\): \(x = 3\) or \(x = -3\).
  • The second quadratic \(-18y^2 = 0\) simplifies to \(y^2 = 0\), indicating a unique solution of \(y = 0\).
This demonstrates how powerful quadratic equations are in providing possible values for variables, and when combined with methods like elimination, allows us to solve complex nonlinear systems effectively.

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Most popular questions from this chapter

For the following exercises, solve the system using the inverse of a \(2 \times 2\) matrix. $$ \begin{array}{l} \frac{1}{2} x+\frac{1}{5} y=-\frac{1}{4} \\ \frac{1}{2} x-\frac{3}{5} y=-\frac{9}{4} \end{array} $$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Men aged \(20-29,30-39,\) and 40-49 made up \(78 \%\) of the population at a prison last year. This year, the same age groups made up \(82.08 \%\) of the population. The \(20-29\) age group increased by \(20 \%,\) the 30-39 age group increased by \(2 \%,\) and the \(40-49\) age group decreased to \(\frac{3}{4}\) of their previous population. Originally, the \(30-39\) age group had \(2 \%\) more prisoners than the \(20-29\) age group. Determine the prison population percentage for each age group last year.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. Three bands performed at a concert venue. The first band charged \(\$ 15\) per ticket, the second band charged \(\$ 45\) per ticket, and the final band charged \(\$ 22\) per ticket. There were 510 tickets sold, for a total of \(\$ 12,700\). If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

For the following exercises, find the determinant. $$ \left|\begin{array}{cc} 10 & 0.2 \\ 5 & 0.1 \end{array}\right| $$

For the following exercises, use a calculator to solve the system of equations with matrix inverses. $$ \begin{array}{l} 2 x-y=-3 \\ -x+2 y=2.3 \end{array} $$
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