Set Up The System Of Equations:A Number Minus Three Times Another Number Is 13. Two Times The First Number Plus Three Times The Second Number Is 28.Set Up The System Of Equations To Find The Numbers:$[ \begin{align*} x - 3y &= 13 \ 2x + 3y &=

by ADMIN 243 views

Introduction

Systems of equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, economics, and computer science. In this article, we will focus on setting up and solving a system of linear equations using two variables. We will use a real-world example to illustrate the process.

Problem Statement

We are given two equations:

  1. A number minus three times another number is 13.
  2. Two times the first number plus three times the second number is 28.

Our goal is to find the values of the two numbers that satisfy both equations.

Setting Up the System of Equations

Let's denote the first number as x and the second number as y. We can now set up the system of equations based on the given information.

Equation 1: A number minus three times another number is 13

We can write this equation as:

x - 3y = 13

Equation 2: Two times the first number plus three times the second number is 28

We can write this equation as:

2x + 3y = 28

The System of Equations

Now that we have set up the two equations, we can write the system of equations as:

x - 3y = 13 ... (Equation 1) 2x + 3y = 28 ... (Equation 2)

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation 1 by 2

We will multiply Equation 1 by 2 to make the coefficients of x in both equations the same.

2(x - 3y) = 2(13) 2x - 6y = 26

Step 2: Add the two equations

We will add the two equations to eliminate the variable x.

(2x - 6y) + (2x + 3y) = 26 + 28 4x - 3y = 54

Step 3: Solve for x

We will solve for x by isolating it on one side of the equation.

4x = 54 + 3y 4x = 54 + 3y x = (54 + 3y) / 4

Step 4: Substitute x into Equation 1

We will substitute x into Equation 1 to solve for y.

((54 + 3y) / 4) - 3y = 13 54 + 3y - 12y = 52 -9y = -2 y = 2/9

Step 5: Substitute y into Equation 1

We will substitute y into Equation 1 to solve for x.

x - 3(2/9) = 13 x - 2/3 = 13 x = 13 + 2/3 x = 39/3 + 2/3 x = 41/3

Conclusion

In this article, we set up and solved a system of linear equations using two variables. We used the elimination method to solve the system and found the values of the two numbers that satisfy both equations. The first number is x = 41/3, and the second number is y = 2/9.

Real-World Applications

Systems of equations have numerous applications in various fields, including:

  • Physics: To solve problems involving motion, forces, and energies.
  • Engineering: To design and optimize systems, such as electrical circuits, mechanical systems, and control systems.
  • Economics: To model and analyze economic systems, including supply and demand, production, and consumption.
  • Computer Science: To solve problems involving algorithms, data structures, and computer networks.

Final Thoughts

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables. In this article, we focused on systems of linear equations with two variables.

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

A: To determine if a system of equations has a solution, you can use the following methods:

  • Graphical method: Graph the two equations on a coordinate plane. If the lines intersect, the system has a solution.
  • Substitution method: Solve one equation for one variable and substitute the expression into the other equation. If the resulting equation is true, the system has a solution.
  • Elimination method: Add or subtract the two equations to eliminate one variable. If the resulting equation is true, the system has a solution.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not linear, meaning they do not have a constant slope.

Q: How do I solve a system of nonlinear equations?

A: To solve a system of nonlinear equations, you can use the following methods:

  • Graphical method: Graph the two equations on a coordinate plane. If the curves intersect, the system has a solution.
  • Numerical method: Use numerical methods, such as the Newton-Raphson method, to approximate the solution.
  • Analytical method: Use analytical methods, such as substitution or elimination, to solve the system.

Q: What is the importance of systems of equations in real-world applications?

A: Systems of equations have numerous applications in various fields, including:

  • Physics: To solve problems involving motion, forces, and energies.
  • Engineering: To design and optimize systems, such as electrical circuits, mechanical systems, and control systems.
  • Economics: To model and analyze economic systems, including supply and demand, production, and consumption.
  • Computer Science: To solve problems involving algorithms, data structures, and computer networks.

Q: How do I check my solution to a system of equations?

A: To check your solution to a system of equations, you can substitute the values of the variables back into the original equations. If the resulting equations are true, the solution is correct.

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 solution: Failing to check the solution by substituting the values of the variables back into the original equations.
  • Not using the correct method: Using the wrong method to solve the system, such as using the substitution method when the elimination method is more efficient.
  • Not being careful with signs: Failing to be careful with signs when adding or subtracting equations.

Conclusion

In this article, we answered some frequently asked questions about systems of equations. We covered topics such as the definition of a system of equations, how to determine if a system has a solution, and how to solve systems of linear and nonlinear equations. We also discussed the importance of systems of equations in real-world applications and provided tips for checking solutions and avoiding common mistakes.