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Introduction

Solving a system of equations algebraically involves using mathematical techniques to find the values of variables that satisfy all the equations in the system. In this article, we will focus on solving a specific system of equations using algebraic methods. We will use the given system of equations:

$$\begin{align*} y + x &= 19 - x^2 \\ x + y &= 80 \end{align*}$$

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

Step 1: Isolate One Variable in the System of Equations

To start solving the system of equations, we need to isolate one variable in one of the equations. Let's isolate $y$ in the second equation:

$$x + y = 80$$

Subtracting $x$ from both sides gives us:

$$y = 80 - x$$

Now we have isolated $y$ in terms of $x$.

Step 2: Use Substitution to Create a One-Variable Equation

We can substitute the expression for $y$ from the second equation into the first equation:

$$y + x = 19 - x^2$$

Substituting $y = 80 - x$ into the first equation gives us:

$$(80 - x) + x = 19 - x^2$$

Simplifying the equation gives us:

$$80 = 19 - x^2$$

Subtracting 19 from both sides gives us:

$$61 = -x^2$$

Multiplying both sides by -1 gives us:

$$-61 = x^2$$

Taking the square root of both sides gives us:

$$x = \pm \sqrt{-61}$$

However, since the square of any real number is non-negative, there is no real solution for $x$. This means that the system of equations has no real solution.

Conclusion

In this article, we have shown how to solve a system of equations algebraically using substitution. We isolated one variable in one of the equations and then substituted the expression into the other equation. However, in this case, the system of equations has no real solution. This is because the equation $x^2 = -61$ has no real solution.

Discussion

The system of equations we solved in this article is a quadratic system of equations. Quadratic systems of equations are systems of equations that involve quadratic expressions. Solving quadratic systems of equations can be challenging, and in some cases, there may not be a real solution.

Tips and Tricks

When solving a system of equations algebraically, it's essential to isolate one variable in one of the equations. This will make it easier to substitute the expression into the other equation. Additionally, when working with quadratic expressions, it's crucial to remember that the square of any real number is non-negative.

Example Problems

Here are some example problems that involve solving systems of equations algebraically:

1. Solve the system of equations:

$$\begin{align*} x + y &= 10 \\ y - x &= 2 \end{align*}$$

2. Solve the system of equations:

$$\begin{align*} x^2 + y^2 &= 25 \\ x + y &= 5 \end{align*}$$

Glossary

• System of equations: A set of equations that involve multiple variables.
• Quadratic expression: An expression that involves a squared variable.
• Isolate: To solve for one variable in an equation.
• Substitution: A method of solving a system of equations by substituting an expression into another equation.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Introduction

Solving a system of equations algebraically can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving a system of equations algebraically.

Q: What is a system of equations?

A: A system of equations is a set of equations that involve multiple variables. Each equation in the system is a statement that two or more expressions are equal.

Q: What are the different methods of solving a system of equations?

A: There are several methods of solving a system of equations, including:

• Substitution method: This method involves substituting an expression into another equation to solve for one variable.
• Elimination method: This method involves adding or subtracting equations to eliminate one variable.
• Graphical method: This method involves graphing the equations on a coordinate plane to find the point of intersection.
• Algebraic method: This method involves using algebraic techniques, such as factoring and expanding, to solve for one variable.

Q: What is the substitution method?

A: The substitution method involves substituting an expression into another equation to solve for one variable. This method is useful when one equation is already solved for one variable.

Q: How do I use the substitution method?

A: To use the substitution method, follow these steps:

1. Solve one equation for one variable.
2. Substitute the expression into the other equation.
3. Solve for the other variable.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting equations to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Q: How do I use the elimination method?

A: To use the elimination method, follow these steps:

1. Add or subtract the equations to eliminate one variable.
2. Solve for the other variable.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane to find the point of intersection. This method is useful when the equations are linear.

Q: How do I use the graphical method?

A: To use the graphical method, follow these steps:

1. Graph the equations on a coordinate plane.
2. Find the point of intersection.

Q: What is the algebraic method?

A: The algebraic method involves using algebraic techniques, such as factoring and expanding, to solve for one variable. This method is useful when the equations are quadratic.

Q: How do I use the algebraic method?

A: To use the algebraic method, follow these steps:

1. Factor or expand the equations.
2. Solve for one variable.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

• Not checking the solution: Make sure to check the solution to ensure that it satisfies both equations.
• Not using the correct method: Choose the correct method for the type of equations you are working with.
• Not following the steps: Follow the steps carefully to ensure that you are solving the system correctly.

Conclusion

Solving a system of equations algebraically can be a challenging task, but with the right techniques and strategies, it can be made easier. By following the steps outlined in this article, you can learn how to solve a system of equations using the substitution, elimination, graphical, and algebraic methods. Remember to check your solution and avoid common mistakes to ensure that you are solving the system correctly.

Glossary

• System of equations: A set of equations that involve multiple variables.
• Substitution method: A method of solving a system of equations by substituting an expression into another equation.
• Elimination method: A method of solving a system of equations by adding or subtracting equations to eliminate one variable.
• Graphical method: A method of solving a system of equations by graphing the equations on a coordinate plane.
• Algebraic method: A method of solving a system of equations by using algebraic techniques, such as factoring and expanding.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart

Note: The references provided are for example purposes only and may not be actual references used in the article.