Solving Linear-Quadratic SystemsAssignmentSolving A System Of Equations Using SubstitutionSolve This System Of Equations Algebraically:${ \begin{array}{l} Y + X = 19 - X^2 \ X + Y = 80 \end{array} }$1. Isolate One Variable In The System Of

Introduction

Solving linear-quadratic systems is a fundamental concept in algebra that involves finding the solution to a system of equations where one equation is linear and the other is quadratic. In this article, we will explore the different methods of solving linear-quadratic systems, including substitution and elimination. We will also provide step-by-step examples to help illustrate the concepts.

What are Linear-Quadratic Systems?

A linear-quadratic system is a system of two equations where one equation is linear and the other is quadratic. A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example:

${ \begin{array}{l} y + x = 19 - x^2 \\ x + y = 80 \end{array} }$

In this example, the first equation is quadratic because it contains a term with a power of 2, while the second equation is linear because it does not contain any terms with a power greater than 1.

Method 1: Substitution

One method of solving linear-quadratic systems is by substitution. This involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Step 1: Isolate one variable in the system of equations

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

${ x + y = 80 }$

Subtracting x from both sides gives us:

${ y = 80 - x }$

Step 2: Substitute the expression into the other equation

Now that we have isolated the variable y, we can substitute this expression into the first equation:

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

Substituting y = 80 - x into this equation gives us:

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

Simplifying this equation gives us:

${ 80 = 19 - x^2 }$

Step 3: Solve for the variable

Now that we have a quadratic equation, we can solve for the variable x. To do this, we need to isolate the variable x. 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 to this equation.

Conclusion

In this example, we were unable to find a real solution to the system of equations using the substitution method. This is because the quadratic equation that we obtained had no real solutions.

Method 2: Elimination

Another method of solving linear-quadratic systems is by elimination. This involves adding or subtracting the two equations to eliminate one of the variables.

Step 1: Add the two equations

To begin, we can add the two equations together:

${ (y + x) + (x + y) = (19 - x^2) + 80 }$

Simplifying this equation gives us:

${ 2x + 2y = 99 - x^2 }$

Step 2: Divide both sides by 2

Dividing both sides of this equation by 2 gives us:

${ x + y = 49.5 - 0.5x^2 }$

Step 3: Solve for the variable

Now that we have a quadratic equation, we can solve for the variable x. To do this, we need to isolate the variable x. Subtracting 49.5 from both sides gives us:

${ x + y - 49.5 = -0.5x^2 }$

However, this equation is not in the standard form of a quadratic equation, and we cannot solve for x using this method.

Conclusion

In this example, we were unable to find a real solution to the system of equations using the elimination method. This is because the quadratic equation that we obtained was not in the standard form.

Solving a System of Equations Using Substitution

Let's consider another example of a system of equations:

${ \begin{array}{l} y + x = 19 - x^2 \\ x + y = 80 \end{array} }$

To solve this system of equations using substitution, we can follow the same steps as before:

Step 1: Isolate one variable in the system of equations

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

${ x + y = 80 }$

Subtracting x from both sides gives us:

${ y = 80 - x }$

Step 2: Substitute the expression into the other equation

Now that we have isolated the variable y, we can substitute this expression into the first equation:

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

Substituting y = 80 - x into this equation gives us:

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

Simplifying this equation gives us:

${ 80 = 19 - x^2 }$

Step 3: Solve for the variable

Now that we have a quadratic equation, we can solve for the variable x. To do this, we need to isolate the variable x. 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 to this equation.

Conclusion

In this example, we were unable to find a real solution to the system of equations using the substitution method. This is because the quadratic equation that we obtained had no real solutions.

Conclusion

Solving linear-quadratic systems is a fundamental concept in algebra that involves finding the solution to a system of equations where one equation is linear and the other is quadratic. In this article, we have explored the different methods of solving linear-quadratic systems, including substitution and elimination. We have also provided step-by-step examples to help illustrate the concepts. However, in both examples, we were unable to find a real solution to the system of equations using either the substitution or elimination method. This is because the quadratic equations that we obtained had no real solutions.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Glossary

• Linear equation: An equation in which the highest power of the variable is 1.
• Quadratic equation: An equation in which the highest power of the variable is 2.
• Substitution method: A method of solving a system of equations by substituting one expression into another.
• Elimination method: A method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables.
  Solving Linear-Quadratic Systems: A Q&A Guide =====================================================

Introduction

Solving linear-quadratic systems is a fundamental concept in algebra that involves finding the solution to a system of equations where one equation is linear and the other is quadratic. In this article, we will provide a Q&A guide to help you understand the concepts and methods of solving linear-quadratic systems.

Q: What is a linear-quadratic system?

A: A linear-quadratic system is a system of two equations where one equation is linear and the other is quadratic. A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear-quadratic system?

A: There are two main methods of solving linear-quadratic systems: substitution and elimination. The substitution method involves solving one of the equations for one of the variables and then substituting that expression into the other equation. The elimination method involves adding or subtracting the two equations to eliminate one of the variables.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by substituting one expression into another. To use the substitution method, you need to isolate one of the variables in one of the equations and then substitute that expression into the other equation.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables. To use the elimination method, you need to add or subtract the two equations in such a way that one of the variables is eliminated.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations. If one of the equations is easily solvable, you may want to use the substitution method. If the equations are more complex, you may want to use the elimination method.

Q: What are some common mistakes to avoid when solving linear-quadratic systems?

A: Some common mistakes to avoid when solving linear-quadratic systems include:

• Not isolating one of the variables in one of the equations
• Not substituting the expression correctly
• Not adding or subtracting the equations correctly
• Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into both of the original equations and check if it satisfies both equations.

Q: What are some real-world applications of linear-quadratic systems?

A: Linear-quadratic systems have many real-world applications, including:

• Physics: to model the motion of objects under the influence of gravity
• Engineering: to design and optimize systems
• Economics: to model the behavior of economic systems

Q: Can I use technology to solve linear-quadratic systems?

A: Yes, you can use technology to solve linear-quadratic systems. Many graphing calculators and computer algebra systems can solve linear-quadratic systems quickly and accurately.

Conclusion

Solving linear-quadratic systems is a fundamental concept in algebra that involves finding the solution to a system of equations where one equation is linear and the other is quadratic. In this article, we have provided a Q&A guide to help you understand the concepts and methods of solving linear-quadratic systems. We hope that this guide has been helpful in your studies.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Glossary

• Linear equation: An equation in which the highest power of the variable is 1.
• Quadratic equation: An equation in which the highest power of the variable is 2.
• Substitution method: A method of solving a system of equations by substituting one expression into another.
• Elimination method: A method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables.