The Sum Of Two Numbers Is 28. The First Number, $x$, Is Three Times The Second Number, $y$.Which System Of Equations Can Be Used To Find The Two Numbers?A. \[$\left\{\begin{array}{l} X + Y = 28 \\ X = 3 + Y

Introduction

In mathematics, a system of equations is a set of two or more equations that are used to solve for the values of variables. In this article, we will explore how to use a system of equations to find the values of two numbers, given that their sum is 28 and the first number is three times the second number.

The Problem

We are given two pieces of information:

1. The sum of the two numbers is 28.
2. The first number, $x$, is three times the second number, $y$.

Setting Up the System of Equations

To set up the system of equations, we need to translate the given information into mathematical equations. Let's start with the first piece of information: the sum of the two numbers is 28.

We can write this as an equation:

$$x + y = 28$$

This equation represents the sum of the two numbers, where $x$ is the first number and $y$ is the second number.

Next, we need to translate the second piece of information: the first number, $x$, is three times the second number, $y$.

We can write this as an equation:

$$x = 3y$$

This equation represents the relationship between the two numbers, where $x$ is three times $y$.

The System of Equations

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

$$\left\{\begin{array}{l} x + y = 28 \\ x = 3y \end{array}\right.$$

This system of equations represents the two pieces of information given in the problem.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.

We can substitute the second equation into the first equation:

$$3y + y = 28$$

Combine like terms:

$$4y = 28$$

Divide both sides by 4:

$$y = 7$$

Now that we have found the value of $y$, we can substitute it into one of the original equations to find the value of $x$.

Let's substitute $y = 7$ into the second equation:

$$x = 3(7)$$

Multiply:

$$x = 21$$

Therefore, the two numbers are $x = 21$ and $y = 7$.

Conclusion

In this article, we have seen how to use a system of equations to find the values of two numbers, given that their sum is 28 and the first number is three times the second number. We have set up the system of equations, solved the system using the method of substitution, and found the values of the two numbers.

Discussion

This problem is a classic example of a system of linear equations. The system of equations can be represented graphically as two lines intersecting at a point, which represents the solution to the system.

The method of substitution is a common technique used to solve systems of linear equations. It involves substituting one equation into another equation to eliminate one of the variables.

In this problem, we have used the method of substitution to solve the system of equations. However, other methods such as the method of elimination or the method of matrices can also be used to solve the system.

Applications

Systems of equations have many applications in mathematics and other fields. Some examples include:

• Physics: Systems of equations are used to model the motion of objects in physics.
• Engineering: Systems of equations are used to design and optimize systems in engineering.
• Economics: Systems of equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Systems of equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Final Thoughts

In conclusion, systems of equations are a powerful tool for solving problems in mathematics and other fields. By using the method of substitution, we can solve systems of linear equations and find the values of variables. The applications of systems of equations are vast and varied, and they have many uses in fields such as physics, engineering, economics, and computer science.

References

• [1] "Systems of Equations" by Khan Academy
• [2] "Linear Equations" by Math Open Reference
• [3] "Systems of Linear Equations" by Wolfram MathWorld

Further Reading

• "Systems of Equations" by MIT OpenCourseWare
• "Linear Algebra" by Stanford University
• "Mathematics for Computer Science" by Harvard University

Related Topics

• Linear Equations: Linear equations are equations in which the highest power of the variable is 1.
• Matrices: Matrices are rectangular arrays of numbers used to represent systems of equations.
• Graphing: Graphing is the process of representing a system of equations as a graph on a coordinate plane.
  

Introduction

Systems of equations are a fundamental concept in mathematics, and they have many applications in various fields. In this article, we will answer some frequently asked questions about systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are used to solve for the values of variables. Each equation in the system is a statement that two expressions are equal.

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

A: You have a system of equations if you have two or more equations that involve the same variables. For example, if you have two equations:

$$x + y = 28$$

$$x = 3y$$

Then you have a system of equations.

Q: What are the different types of systems of equations?

A: There are two main types of systems of equations:

• Linear systems of equations: These are systems of equations in which the highest power of the variable is 1. For example:

$$x + y = 28$$

$$x = 3y$$

• Nonlinear systems of equations: These are systems of equations in which the highest power of the variable is greater than 1. For example:

$$x^2 + y^2 = 28$$

$$x = 3y$$

Q: How do I solve a system of equations?

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

• Substitution method: This involves substituting one equation into another equation to eliminate one of the variables.
• Elimination method: This involves adding or subtracting equations to eliminate one of the variables.
• Graphing method: This involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting one equation into another equation to eliminate one of the variables. For example, if you have two equations:

$$x + y = 28$$

$$x = 3y$$

You can substitute the second equation into the first equation to get:

$$3y + y = 28$$

Combine like terms:

$$4y = 28$$

Divide both sides by 4:

$$y = 7$$

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of equations by adding or subtracting equations to eliminate one of the variables. For example, if you have two equations:

$$x + y = 28$$

$$x = 3y$$

You can add the two equations to get:

$$2x + 4y = 56$$

Subtract the second equation from the first equation to get:

$$-2y = -28$$

Divide both sides by -2:

$$y = 14$$

Q: What is the graphing method?

A: The graphing method is a technique used to solve a system of equations by graphing the equations on a coordinate plane and finding the point of intersection. For example, if you have two equations:

$$x + y = 28$$

$$x = 3y$$

You can graph the two equations on a coordinate plane and find the point of intersection, which is the solution to the system.

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: Make sure to check the solution to the system by plugging it back into the original equations.
• Not using the correct method: Make sure to use the correct method to solve the system, such as substitution or elimination.
• Not simplifying the equations: Make sure to simplify the equations before solving the system.

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

A: Systems of equations have many real-world applications, including:

• Physics: Systems of equations are used to model the motion of objects in physics.
• Engineering: Systems of equations are used to design and optimize systems in engineering.
• Economics: Systems of equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Systems of equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by:

• Solving problems: Practice solving problems that involve systems of equations.
• Using online resources: Use online resources such as Khan Academy, Math Open Reference, and Wolfram MathWorld to practice solving systems of equations.
• Working with a tutor: Work with a tutor to practice solving systems of equations.

Conclusion

In this article, we have answered some frequently asked questions about systems of equations. We have discussed the different types of systems of equations, the methods used to solve them, and some common mistakes to avoid. We have also discussed some real-world applications of systems of equations and provided some tips for practicing solving systems of equations.