Given The System Of Equations:$\[ \begin{align*} y &= -5x + 30 \\ x &= 10 \end{align*} \\]What Is The Solution To The System Of Equations?A. \[$(-20, 10)\$\] B. \[$(10, -20)\$\] C. \[$(10, 4)\$\] D. \[$(4,

by ADMIN 209 views

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. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations to demonstrate the steps involved in solving a system of equations.

The System of Equations

The given system of equations is:

y=−5x+30x=10\begin{align*} y &= -5x + 30 \\ x &= 10 \end{align*}

Understanding the Equations

The first equation is a linear equation in the form of y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. In this case, the slope is −5-5 and the y-intercept is 3030. The second equation is a simple equation that states xx is equal to 1010.

Substituting the Value of x

Since we are given the value of xx as 1010, we can substitute this value into the first equation to find the value of yy. Substituting x=10x = 10 into the first equation, we get:

y=−5(10)+30y = -5(10) + 30

Simplifying the Equation

To simplify the equation, we can multiply −5-5 by 1010 to get −50-50. Then, we can add 3030 to get:

y=−50+30y = -50 + 30

y=−20y = -20

The Solution to the System of Equations

Therefore, the solution to the system of equations is (10,−20)(10, -20).

Conclusion

In this article, we have demonstrated the steps involved in solving a system of two linear equations with two variables. We have used the given system of equations to find the values of the variables. The solution to the system of equations is (10,−20)(10, -20).

Why is this the Correct Solution?

To verify that the solution is correct, we can substitute the values of xx and yy back into the original equations. Substituting x=10x = 10 and y=−20y = -20 into the first equation, we get:

−20=−5(10)+30-20 = -5(10) + 30

−20=−50+30-20 = -50 + 30

−20=−20-20 = -20

This shows that the solution satisfies both equations, and therefore, it is the correct solution.

What is the Importance of Solving Systems of Equations?

Solving systems of equations is an important concept in mathematics and has many real-world applications. It is used in various fields such as physics, engineering, economics, and computer science. For example, in physics, systems of equations are used to describe the motion of objects under the influence of forces. In engineering, systems of equations are used to design and analyze complex systems such as bridges and buildings.

Common Mistakes to Avoid

When solving systems of equations, there are several common mistakes to avoid. These include:

  • Not checking the solution: It is essential to verify that the solution satisfies both equations.
  • Not using the correct method: There are several methods to solve systems of equations, and it is essential to use the correct method for the given system.
  • Not being careful with signs: When simplifying equations, it is essential to be careful with signs to avoid errors.

Conclusion

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: What are the different types of systems of equations?

A: There are two main types of systems of equations: linear systems and nonlinear systems. Linear systems consist of linear equations, while nonlinear systems consist of nonlinear equations.

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

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

  • Substitution method: Substitute the value of one variable from one equation into the other equation.
  • Elimination method: Add or subtract the equations to eliminate one variable.
  • Graphical method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves substituting the value of one variable from one equation into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I know which method to use?

A: The choice of method depends on the type of system and the variables involved. If the system has two variables and two equations, the substitution method or elimination method may be used. If the system has more than two variables or equations, other methods such as matrices or graphing may be used.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful for systems with two variables and two equations.

Q: How do I graph a system of equations?

A: To graph a system of equations, follow these steps:

  1. Graph each equation separately: Graph each equation on a coordinate plane.
  2. Find the point of intersection: Find the point where the two graphs intersect.
  3. Check the solution: Verify that the solution satisfies both equations.

Q: What is the importance of checking the solution?

A: Checking the solution is essential to ensure that the solution satisfies both equations. If the solution does not satisfy one or both equations, it is not a valid solution.

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 verify that the solution satisfies both equations.
  • Not using the correct method: Using the wrong method for the given system.
  • Not being careful with signs: Making errors when simplifying equations.

Q: How do I apply systems of equations to real-world problems?

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

  • Physics: Systems of equations are used to describe the motion of objects under the influence of forces.
  • Engineering: Systems of equations are used to design and analyze complex systems such as bridges and buildings.
  • Economics: Systems of equations are used to model economic systems and make predictions about economic trends.

Conclusion

In conclusion, solving systems of equations is an important concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve systems of equations and apply the concepts to real-world problems. Remember to verify the solution, use the correct method, and be careful with signs to avoid errors.