Solve The Following System Of Equations:${ \begin{aligned} x^2 + Y^2 &= 25 \ y - 2x &= 5 \end{aligned} }$

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Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two equations with two unknowns, which is a common scenario in algebra and geometry. We will use a step-by-step approach to solve the system of equations and provide a detailed explanation of each step.

The System of Equations

The system of equations we will be solving is:

x2+y2=25y−2x=5\begin{aligned} x^2 + y^2 &= 25 \\ y - 2x &= 5 \end{aligned}

Step 1: Understand the Equations

The first equation is a quadratic equation in two variables, x and y. It represents a circle with a radius of 5 units, centered at the origin (0, 0). The second equation is a linear equation in two variables, x and y. It represents a line with a slope of 2 and a y-intercept of 5.

Step 2: Solve the Linear Equation for y

We can solve the linear equation for y by isolating y on one side of the equation. We get:

y=2x+5y = 2x + 5

Step 3: Substitute the Expression for y into the Quadratic Equation

We can substitute the expression for y into the quadratic equation by replacing y with 2x + 5. We get:

x2+(2x+5)2=25x^2 + (2x + 5)^2 = 25

Step 4: Expand and Simplify the Equation

We can expand and simplify the equation by multiplying out the squared term and combining like terms. We get:

x2+4x2+20x+25=25x^2 + 4x^2 + 20x + 25 = 25

Step 5: Combine Like Terms

We can combine like terms by adding or subtracting coefficients of similar terms. We get:

5x2+20x=05x^2 + 20x = 0

Step 6: Factor Out the Greatest Common Factor

We can factor out the greatest common factor (GCF) of the terms by identifying the largest factor that divides both terms. We get:

5x(x+4)=05x(x + 4) = 0

Step 7: Solve for x

We can solve for x by setting each factor equal to zero and solving for x. We get:

x=0orx+4=0x = 0 \quad \text{or} \quad x + 4 = 0

Step 8: Solve for x + 4 = 0

We can solve for x + 4 = 0 by subtracting 4 from both sides of the equation. We get:

x=−4x = -4

Step 9: Find the Corresponding Values of y

We can find the corresponding values of y by substituting the values of x into the expression for y. We get:

y=2(0)+5=5ory=2(−4)+5=−3y = 2(0) + 5 = 5 \quad \text{or} \quad y = 2(-4) + 5 = -3

Step 10: Check the Solutions

We can check the solutions by substituting the values of x and y into the original equations to ensure that they satisfy both equations.

Conclusion

In this article, we solved a system of two equations with two unknowns using a step-by-step approach. We used algebraic techniques to solve the system of equations and found two solutions: (0, 5) and (-4, -3). We also checked the solutions to ensure that they satisfy both equations.

Final Answer

The final answer is (0,5),(−4,−3)\boxed{(0, 5), (-4, -3)}.

Discussion

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we used a step-by-step approach to solve a system of two equations with two unknowns. We provided a detailed explanation of each step and checked the solutions to ensure that they satisfy both equations.

Related Topics

  • Solving systems of linear equations
  • Solving systems of quadratic equations
  • Algebraic techniques for solving systems of equations
  • Geometric interpretation of systems of equations

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Keywords

  • System of equations
  • Algebra
  • Geometry
  • Linear equations
  • Quadratic equations
  • Algebraic techniques
  • Geometric interpretation

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Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In our previous article, we provided a step-by-step guide on how to solve a system of two equations with two unknowns. In this article, we will answer some frequently asked questions (FAQs) related to solving systems of equations.

Q&A

Q: What is a system of equations?

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

Q: How do I know if a system of equations has a solution?

A: A system of equations has a solution if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution.

Q: What are the different methods for solving systems of equations?

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

  • Substitution method: This method involves substituting the expression for one variable into the other equation.
  • Elimination method: This method involves adding or subtracting the two equations to eliminate one of the variables.
  • Graphical method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
  • Algebraic method: This method involves using algebraic techniques, such as factoring and solving quadratic equations, to solve the system.

Q: How do I choose the best method for solving a system of equations?

A: The best method for solving a system of equations depends on the type of equations and the variables involved. For example, if the equations are linear, the elimination method may be the best choice. If the equations are quadratic, the algebraic method may be the best choice.

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 the consistency of the equations
  • Not using the correct method for the type of equations
  • Not solving for all variables
  • Not checking the solutions for consistency

Q: How do I check the solutions for consistency?

A: To check the solutions for consistency, substitute the values of the variables into the original equations and check if they satisfy both equations.

Q: What are some real-world applications of solving systems of equations?

A: Solving systems of equations has many real-world applications, including:

  • Physics: Solving systems of equations is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Solving systems of equations is used to model economic systems and make predictions about the behavior of markets.

Conclusion

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we answered some frequently asked questions (FAQs) related to solving systems of equations. We hope that this article has provided you with a better understanding of how to solve systems of equations and has helped you to avoid common mistakes.

Final Answer

The final answer is Thereisnofinalanswer,assolvingsystemsofequationsisaprocessthatrequiresastep−by−stepapproach.\boxed{There is no final answer, as solving systems of equations is a process that requires a step-by-step approach.}.

Discussion

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we provided a Q&A guide on how to solve systems of equations. We hope that this article has provided you with a better understanding of how to solve systems of equations and has helped you to avoid common mistakes.

Related Topics

  • Solving systems of linear equations
  • Solving systems of quadratic equations
  • Algebraic techniques for solving systems of equations
  • Geometric interpretation of systems of equations

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Keywords

  • System of equations
  • Algebra
  • Geometry
  • Linear equations
  • Quadratic equations
  • Algebraic techniques
  • Geometric interpretation
  • Real-world applications
  • Physics
  • Engineering
  • Economics