Solve The System Of Equations.${ \begin{array}{l} 2.5y + 3x = 27 \ 5x - 2.5y = 5 \end{array} }$1. What Equation Is The Result Of Adding The Two Equations? { \boxed{\phantom{solution}}$}$2. What Is The Solution To The System?

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Introduction

Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of addition to eliminate one of the variables and then solve for the other variable.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 2.5y + 3x = 27 \\ 5x - 2.5y = 5 \end{array} \}

Adding the Two Equations

To add the two equations, we need to add the corresponding terms. The first equation has a term of 2.5y, and the second equation has a term of -2.5y. When we add these two terms, they cancel each other out. The first equation has a term of 3x, and the second equation has a term of 5x. When we add these two terms, we get 8x. The first equation has a constant term of 27, and the second equation has a constant term of 5. When we add these two terms, we get 32.

The result of adding the two equations is:

8x=32{ 8x = 32 }

Solving for x

To solve for x, we need to isolate x on one side of the equation. We can do this by dividing both sides of the equation by 8.

x=328{ x = \frac{32}{8} }

x=4{ x = 4 }

Substituting x into One of the Original Equations

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the first equation:

2.5y+3x=27{ 2.5y + 3x = 27 }

Substituting x = 4 into this equation, we get:

2.5y+3(4)=27{ 2.5y + 3(4) = 27 }

2.5y+12=27{ 2.5y + 12 = 27 }

Solving for y

To solve for y, we need to isolate y on one side of the equation. We can do this by subtracting 12 from both sides of the equation.

2.5y=27−12{ 2.5y = 27 - 12 }

2.5y=15{ 2.5y = 15 }

Next, we can divide both sides of the equation by 2.5 to solve for y.

y=152.5{ y = \frac{15}{2.5} }

y=6{ y = 6 }

The Solution to the System

The solution to the system of equations is x = 4 and y = 6.

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of addition. We added the two equations to eliminate one of the variables and then solved for the other variable. We found that the solution to the system is x = 4 and y = 6.

Discussion

Solving a system of linear equations is an important skill in mathematics and is used in a wide range of applications, including science, engineering, and economics. There are many different methods for solving a system of linear equations, including the method of substitution, the method of elimination, and the method of matrices. In this article, we focused on the method of addition, which is a simple and effective way to solve a system of linear equations.

Example Problems

Problem 1

Solve the system of equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 2 \end{array} \}

Solution

To solve this system, we can add the two equations to eliminate one of the variables.

(x+2y)+(3x−2y)=6+2{ (x + 2y) + (3x - 2y) = 6 + 2 }

4x=8{ 4x = 8 }

x=2{ x = 2 }

Substituting x = 2 into one of the original equations, we get:

x+2y=6{ x + 2y = 6 }

2+2y=6{ 2 + 2y = 6 }

2y=4{ 2y = 4 }

y=2{ y = 2 }

The solution to the system is x = 2 and y = 2.

Problem 2

Solve the system of equations:

{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}

Solution

To solve this system, we can add the two equations to eliminate one of the variables.

(2x+3y)+(x−2y)=12+(−3){ (2x + 3y) + (x - 2y) = 12 + (-3) }

3x+y=9{ 3x + y = 9 }

Substituting y = 9 - 3x into one of the original equations, we get:

2x+3y=12{ 2x + 3y = 12 }

2x+3(9−3x)=12{ 2x + 3(9 - 3x) = 12 }

2x+27−9x=12{ 2x + 27 - 9x = 12 }

−7x=−15{ -7x = -15 }

x=−15−7{ x = \frac{-15}{-7} }

x=157{ x = \frac{15}{7} }

Substituting x = 15/7 into the equation 3x + y = 9, we get:

3(157)+y=9{ 3(\frac{15}{7}) + y = 9 }

457+y=9{ \frac{45}{7} + y = 9 }

y=9−457{ y = 9 - \frac{45}{7} }

y=637−457{ y = \frac{63}{7} - \frac{45}{7} }

y=187{ y = \frac{18}{7} }

The solution to the system is x = 15/7 and y = 18/7.

Final Thoughts

Solving a system of linear equations is an important skill in mathematics and is used in a wide range of applications. In this article, we focused on the method of addition, which is a simple and effective way to solve a system of linear equations. We also provided two example problems to help illustrate the method. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to a wide range of real-world problems.

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Introduction

Solving a system of linear equations is an important skill in mathematics and is used in a wide range of applications. In this article, we will provide a Q&A section to help answer some common questions about solving systems of linear equations.

Q&A

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What are the different methods for solving a system of linear equations?

A: There are several methods for solving a system of linear equations, including the method of substitution, the method of elimination, and the method of matrices.

Q: What is the method of substitution?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the method of matrices?

A: The method of matrices involves using matrices to solve the system of equations.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. It's often helpful to try out different methods to see which one works best.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid include:

  • Not following the order of operations
  • Not isolating the variables correctly
  • Not checking the solution for consistency

Q: How do I check the solution for consistency?

A: To check the solution for consistency, substitute the values of the variables back into the original equations and make sure they are true.

Q: What if I get a system of linear equations with no solution?

A: If you get a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

Q: What if I get a system of linear equations with infinitely many solutions?

A: If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.

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

A: To graph a system of linear equations, plot the equations on a coordinate plane and find the point of intersection.

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

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

  • Physics: to solve problems involving motion and forces
  • Engineering: to design and optimize systems
  • Economics: to model and analyze economic systems
  • Computer Science: to solve problems involving algorithms and data structures

Conclusion

Solving a system of linear equations is an important skill in mathematics and has many real-world applications. In this article, we provided a Q&A section to help answer some common questions about solving systems of linear equations. We hope this article has been helpful in providing a better understanding of this important topic.

Final Thoughts

Solving a system of linear equations is a fundamental concept in mathematics and has many real-world applications. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to a wide range of problems.