Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section9.7: Solving Systems Of Linear And Quadratic Equations
Problem 3GP
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![### Solving Systems of Equations by the Elimination Method
In this exercise, we will use the elimination method to find all solutions for the given system of equations. The solutions should be ordered based on the variable \( y \) from smallest to largest.
The system of equations is as follows:
\[
\begin{cases}
3x^2 - y^2 = 8 \\
x^2 + 3y^2 = 16
\end{cases}
\]
Follow these steps to solve this system using the elimination method:
1. **Eliminate one of the variables**: By manipulating the given equations, you can add or subtract them to eliminate either \( x^2 \) or \( y^2 \).
2. **Solve the resulting equation**: Once one variable is eliminated, you will be left with a single-variable equation, which can be solved using conventional methods.
3. **Substitute back**: Substitute the solution(s) from step 2 back into one of the original equations to find the corresponding values of the other variable.
4. **Order the solutions**: Once you have all pairs \((x, y)\), arrange them from the smallest to largest \( y \).
This method is systematic and can be applied to various systems of equations to arrive at the correct solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d7cd53b-4087-4874-ad94-fbe23050be90%2F2d06560f-7ce4-4db1-92ff-9b7320cb9a70%2Fkuihkhe_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Equations by the Elimination Method
In this exercise, we will use the elimination method to find all solutions for the given system of equations. The solutions should be ordered based on the variable \( y \) from smallest to largest.
The system of equations is as follows:
\[
\begin{cases}
3x^2 - y^2 = 8 \\
x^2 + 3y^2 = 16
\end{cases}
\]
Follow these steps to solve this system using the elimination method:
1. **Eliminate one of the variables**: By manipulating the given equations, you can add or subtract them to eliminate either \( x^2 \) or \( y^2 \).
2. **Solve the resulting equation**: Once one variable is eliminated, you will be left with a single-variable equation, which can be solved using conventional methods.
3. **Substitute back**: Substitute the solution(s) from step 2 back into one of the original equations to find the corresponding values of the other variable.
4. **Order the solutions**: Once you have all pairs \((x, y)\), arrange them from the smallest to largest \( y \).
This method is systematic and can be applied to various systems of equations to arrive at the correct solutions.
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