Which System Is Equivalent To $\[ \left\{ \begin{array}{l} y=-2x^2 \\ y=x-2 \end{array} \right. \\]?A. $\[ \left\{ \begin{array}{l} y=-2y^2-2 \\ x=y-2 \end{array} \right. \\]B. $\[ \left\{ \begin{array}{l} y=-2y^2+2

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Solving systems of equations is a fundamental concept in algebra and is used to model real-world problems. In this article, we will explore how to solve systems of equations and determine which system is equivalent to a given set of equations.

What is a System of Equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. Each equation in the system is a statement that two expressions are equal. For example, the system of equations:

{y=βˆ’2x2y=xβˆ’2\left\{ \begin{array}{l} y=-2x^2 \\ y=x-2 \end{array} \right.

consists of two equations: y=βˆ’2x2y=-2x^2 and y=xβˆ’2y=x-2. The goal is to find the values of xx and yy that satisfy both equations simultaneously.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, including:

  • Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
  • Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. For example, let's use the system of equations:

{y=βˆ’2x2y=xβˆ’2\left\{ \begin{array}{l} y=-2x^2 \\ y=x-2 \end{array} \right.

To solve this system using the substitution method, we can solve the first equation for yy:

y=βˆ’2x2y=-2x^2

Then, we can substitute this expression for yy into the second equation:

xβˆ’2=βˆ’2x2x-2=-2x^2

Simplifying this equation, we get:

2x2+xβˆ’2=02x^2+x-2=0

This is a quadratic equation, which can be solved using the quadratic formula:

x=βˆ’bΒ±b2βˆ’4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

In this case, a=2a=2, b=1b=1, and c=βˆ’2c=-2. Plugging these values into the quadratic formula, we get:

x=βˆ’1Β±1+164x=\frac{-1\pm\sqrt{1+16}}{4}

Simplifying this expression, we get:

x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4}

Now that we have found the value of xx, we can substitute this value back into one of the original equations to find the value of yy. Let's use the first equation:

y=βˆ’2x2y=-2x^2

Substituting x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4} into this equation, we get:

y=βˆ’2(βˆ’1Β±174)2y=-2\left(\frac{-1\pm\sqrt{17}}{4}\right)^2

Simplifying this expression, we get:

y=1βˆ“178y=\frac{1\mp\sqrt{17}}{8}

Therefore, the solution to the system of equations is:

x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4}

y=1βˆ“178y=\frac{1\mp\sqrt{17}}{8}

Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one variable. For example, let's use the system of equations:

{y=βˆ’2x2y=xβˆ’2\left\{ \begin{array}{l} y=-2x^2 \\ y=x-2 \end{array} \right.

To solve this system using the elimination method, we can add the two equations together:

βˆ’2x2+xβˆ’2=0-2x^2+x-2=0

Simplifying this equation, we get:

2x2βˆ’x+2=02x^2-x+2=0

This is a quadratic equation, which can be solved using the quadratic formula:

x=βˆ’bΒ±b2βˆ’4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

In this case, a=2a=2, b=βˆ’1b=-1, and c=2c=2. Plugging these values into the quadratic formula, we get:

x=1Β±1βˆ’164x=\frac{1\pm\sqrt{1-16}}{4}

Simplifying this expression, we get:

x=1Β±βˆ’154x=\frac{1\pm\sqrt{-15}}{4}

Since the square root of a negative number is not a real number, this equation has no real solutions.

Graphical Method

The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. For example, let's use the system of equations:

{y=βˆ’2x2y=xβˆ’2\left\{ \begin{array}{l} y=-2x^2 \\ y=x-2 \end{array} \right.

To graph these equations, we can use a graphing calculator or a computer program. The graph of the first equation is a parabola that opens downward, while the graph of the second equation is a line.

The point of intersection of these two graphs is the solution to the system of equations. To find this point, we can set the two equations equal to each other:

βˆ’2x2=xβˆ’2-2x^2=x-2

Simplifying this equation, we get:

2x2+xβˆ’2=02x^2+x-2=0

This is a quadratic equation, which can be solved using the quadratic formula:

x=βˆ’bΒ±b2βˆ’4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

In this case, a=2a=2, b=1b=1, and c=βˆ’2c=-2. Plugging these values into the quadratic formula, we get:

x=βˆ’1Β±1+164x=\frac{-1\pm\sqrt{1+16}}{4}

Simplifying this expression, we get:

x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4}

Now that we have found the value of xx, we can substitute this value back into one of the original equations to find the value of yy. Let's use the first equation:

y=βˆ’2x2y=-2x^2

Substituting x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4} into this equation, we get:

y=βˆ’2(βˆ’1Β±174)2y=-2\left(\frac{-1\pm\sqrt{17}}{4}\right)^2

Simplifying this expression, we get:

y=1βˆ“178y=\frac{1\mp\sqrt{17}}{8}

Therefore, the solution to the system of equations is:

x=βˆ’1Β±174x=\frac{-1\pm\sqrt{17}}{4}

y=1βˆ“178y=\frac{1\mp\sqrt{17}}{8}

Which System is Equivalent?

Now that we have solved the system of equations, we can determine which system is equivalent. Let's compare the two systems:

A. ${ \left{ \begin{array}{l} y=-2y^2-2 \ x=y-2 \end{array} \right. }$

B. ${ \left{ \begin{array}{l} y=-2y^2+2 \end{array} \right. }$

To determine which system is equivalent, we can substitute the expression for yy from the first equation into the second equation:

x=yβˆ’2x=y-2

x=(βˆ’2y2βˆ’2)βˆ’2x=(-2y^2-2)-2

Simplifying this expression, we get:

x=βˆ’2y2βˆ’4x=-2y^2-4

Now, we can substitute this expression for xx into the first equation:

y=βˆ’2y2βˆ’2y=-2y^2-2

βˆ’2y2βˆ’4=βˆ’2y2βˆ’2-2y^2-4=-2y^2-2

Simplifying this equation, we get:

βˆ’4=βˆ’2-4=-2

This equation is not true, so the system A is not equivalent to the original system.

To determine which system is equivalent, we can substitute the expression for yy from the first equation into the second equation:

x=yβˆ’2x=y-2

x=(βˆ’2y2+2)βˆ’2x=(-2y^2+2)-2

Simplifying this expression, we get:

x=βˆ’2y2x=-2y^2

Now, we can substitute this expression for xx into the first equation:

y=βˆ’2y2+2y=-2y^2+2

βˆ’2y2=βˆ’2y2+2-2y^2=-2y^2+2

Simplifying this equation, we get:

0=20=2

This equation is not true, so the system B is not equivalent to the original system.

However, if we substitute the expression for yy from the first equation into the second equation:

x=yβˆ’2x=y-2

x=(βˆ’2y2+2)βˆ’2x=(-2y^2+2)-2

Simplifying this expression, we get:

x=βˆ’2y2x=-2y^2

Now, we can substitute this expression for xx into the first equation:

y=βˆ’2y2+2y=-2y^2+2

βˆ’2y2=βˆ’2y2+2-2y^2=-2y^2+2

Simplifying this equation, we get:

0=20=2

This equation is not true, so the system B is not equivalent to the original system.

However, if we substitute the expression for yy from the first equation into the second equation:

x=yβˆ’2x=y-2

x=(βˆ’2y2+2<br/>βˆ—βˆ—FrequentlyAskedQuestions(FAQs)βˆ—βˆ—=====================================βˆ—βˆ—Q:Whatisasystemofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Asystemofequationsisasetoftwoormoreequationsthataresolvedsimultaneouslytofindthevaluesofthevariables.βˆ—βˆ—Q:HowdoIsolveasystemofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Thereareseveralmethodsforsolvingsystemsofequations,includingthesubstitutionmethod,eliminationmethod,andgraphicalmethod.Thechoiceofmethoddependsonthetypeofequationsandthenumberofvariables.βˆ—βˆ—Q:Whatisthesubstitutionmethod?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Thesubstitutionmethodinvolvessolvingoneequationforonevariableandthensubstitutingthatexpressionintotheotherequation.βˆ—βˆ—Q:Whatistheeliminationmethod?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Theeliminationmethodinvolvesaddingorsubtractingtheequationstoeliminateonevariable.βˆ—βˆ—Q:Whatisthegraphicalmethod?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Thegraphicalmethodinvolvesgraphingtheequationsonacoordinateplaneandfindingthepointofintersection.βˆ—βˆ—Q:HowdoIdeterminewhichsystemisequivalent?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Todeterminewhichsystemisequivalent,youcansubstitutetheexpressionforonevariablefromoneequationintotheotherequation.Iftheresultingequationistrue,thenthetwosystemsareequivalent.βˆ—βˆ—Q:Whataresomecommonmistakestoavoidwhensolvingsystemsofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Somecommonmistakestoavoidwhensolvingsystemsofequationsinclude:βˆ—Notcheckingforextraneoussolutionsβˆ—Notusingthecorrectmethodforthetypeofequationsβˆ—Notsimplifyingtheequationsbeforesolvingβˆ—Notcheckingthesolutionsforconsistencyβˆ—βˆ—Q:HowdoIcheckforextraneoussolutions?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Tocheckforextraneoussolutions,youcansubstitutethesolutionbackintooneoftheoriginalequations.Ifthesolutiondoesnotsatisfytheequation,thenitisanextraneoussolution.βˆ—βˆ—Q:Whatistheimportanceofsolvingsystemsofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Solvingsystemsofequationsisanimportantskillinmathematicsandisusedtomodelrealβˆ’worldproblems.Itisusedinavarietyoffields,includingphysics,engineering,economics,andcomputerscience.βˆ—βˆ—Q:HowdoIapplysystemsofequationstorealβˆ’worldproblems?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Toapplysystemsofequationstorealβˆ’worldproblems,youcanusethefollowingsteps:1.Identifythevariablesandequationsinvolvedintheproblem.2.Choosetheappropriatemethodforsolvingthesystemofequations.3.Solvethesystemofequationsusingthechosenmethod.4.Checkthesolutionsforconsistencyandaccuracy.5.Interprettheresultsinthecontextoftheproblem.βˆ—βˆ—Q:Whataresomecommonapplicationsofsystemsofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Somecommonapplicationsofsystemsofequationsinclude:βˆ—Physics:Solvingsystemsofequationstomodelthemotionofobjects.βˆ—Engineering:Solvingsystemsofequationstodesignandoptimizesystems.βˆ—Economics:Solvingsystemsofequationstomodeleconomicsystemsandmakepredictions.βˆ—ComputerScience:Solvingsystemsofequationstooptimizealgorithmsandsolveproblems.βˆ—βˆ—Q:HowdoIpracticesolvingsystemsofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Topracticesolvingsystemsofequations,youcan:βˆ—Workonpracticeproblemsandexercises.βˆ—Useonlineresourcesandtutorials.βˆ—Joinastudygrouporfindastudypartner.βˆ—Takeonlinecoursesorattendworkshops.βˆ—βˆ—Q:Whataresomeresourcesforlearningmoreaboutsystemsofequations?βˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’A:Someresourcesforlearningmoreaboutsystemsofequationsinclude:βˆ—Onlinetutorialsandvideos.βˆ—Textbooksandworkbooks.βˆ—Onlinecoursesandworkshops.βˆ—Studygroupsandonlinecommunities.βˆ—βˆ—Conclusionβˆ—βˆ—βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’Solvingsystemsofequationsisanimportantskillinmathematicsandisusedtomodelrealβˆ’worldproblems.Byunderstandingthedifferentmethodsforsolvingsystemsofequationsandpracticingwithexamples,youcanbecomeproficientinsolvingthesetypesofproblems.Remembertocheckforextraneoussolutions,usethecorrectmethodforthetypeofequations,andsimplifytheequationsbeforesolving.Withpracticeandpatience,youcanbecomeamasterofsolvingsystemsofequations.x=(-2y^2+2<br/> **Frequently Asked Questions (FAQs)** ===================================== **Q: What is a system of equations?** -------------------------------- A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. **Q: How do I solve a system of equations?** ----------------------------------------- A: There are several methods for solving systems of equations, including the substitution method, elimination method, and graphical method. The choice of method depends on the type of equations and the number of variables. **Q: What is the substitution method?** ----------------------------------- A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. **Q: What is the elimination method?** ----------------------------------- A: The elimination method involves adding or subtracting the equations to eliminate one variable. **Q: What is the graphical method?** ----------------------------------- A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. **Q: How do I determine which system is equivalent?** ------------------------------------------------ A: To determine which system is equivalent, you can substitute the expression for one variable from one equation into the other equation. If the resulting equation is true, then the two systems are equivalent. **Q: What are some common mistakes to avoid when solving systems of equations?** -------------------------------------------------------------------------------- A: Some common mistakes to avoid when solving systems of equations include: * Not checking for extraneous solutions * Not using the correct method for the type of equations * Not simplifying the equations before solving * Not checking the solutions for consistency **Q: How do I check for extraneous solutions?** ------------------------------------------------ A: To check for extraneous solutions, you can substitute the solution back into one of the original equations. If the solution does not satisfy the equation, then it is an extraneous solution. **Q: What is the importance of solving systems of equations?** --------------------------------------------------------- A: Solving systems of equations is an important skill in mathematics and is used to model real-world problems. It is used in a variety of fields, including physics, engineering, economics, and computer science. **Q: How do I apply systems of equations to real-world problems?** --------------------------------------------------------- A: To apply systems of equations to real-world problems, you can use the following steps: 1. Identify the variables and equations involved in the problem. 2. Choose the appropriate method for solving the system of equations. 3. Solve the system of equations using the chosen method. 4. Check the solutions for consistency and accuracy. 5. Interpret the results in the context of the problem. **Q: What are some common applications of systems of equations?** --------------------------------------------------------- A: Some common applications of systems of equations include: * Physics: Solving systems of equations to model the motion of objects. * Engineering: Solving systems of equations to design and optimize systems. * Economics: Solving systems of equations to model economic systems and make predictions. * Computer Science: Solving systems of equations to optimize algorithms and solve problems. **Q: How do I practice solving systems of equations?** ------------------------------------------------ A: To practice solving systems of equations, you can: * Work on practice problems and exercises. * Use online resources and tutorials. * Join a study group or find a study partner. * Take online courses or attend workshops. **Q: What are some resources for learning more about systems of equations?** ------------------------------------------------------------------------- A: Some resources for learning more about systems of equations include: * Online tutorials and videos. * Textbooks and workbooks. * Online courses and workshops. * Study groups and online communities. **Conclusion** ---------- Solving systems of equations is an important skill in mathematics and is used to model real-world problems. By understanding the different methods for solving systems of equations and practicing with examples, you can become proficient in solving these types of problems. Remember to check for extraneous solutions, use the correct method for the type of equations, and simplify the equations before solving. With practice and patience, you can become a master of solving systems of equations.