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{} \\]2. What Is The Solution To The System?$\[ \boxed{} \\]

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Introduction

In mathematics, 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. 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. We can do this by adding the coefficients of the variables and the constants.

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

Using the distributive property, we can simplify the equation:

{ \begin{array}{l} (2.5y + 3x) + (5x - 2.5y) = 27 + 5 \\ 3x + 5x = 32 \\ 8x = 32 \\ \end{array} \}

Solving for x

Now that we have eliminated the variable y, we can solve for x. To do this, we need to isolate x by dividing both sides of the equation by 8.

{ \begin{array}{l} 8x = 32 \\ x = \frac{32}{8} \\ x = 4 \\ \end{array} \}

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:

{ \begin{array}{l} 2.5y + 3x = 27 \\ 2.5y + 3(4) = 27 \\ 2.5y + 12 = 27 \\ \end{array} \}

Solving for y

Now that we have substituted x into the equation, we can solve for y. To do this, we need to isolate y by subtracting 12 from both sides of the equation and then dividing both sides by 2.5.

{ \begin{array}{l} 2.5y + 12 = 27 \\ 2.5y = 15 \\ y = \frac{15}{2.5} \\ y = 6 \\ \end{array} \}

The Solution to the System

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

Conclusion

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

Discussion

This system of equations can be solved using other methods such as substitution or elimination. The method of addition is a simple and efficient way to solve systems of linear equations.

Example Problems

  1. Solve the system of equations:

{ \begin{array}{l} x + 2y = 7 \\ 3x - 2y = 11 \\ \end{array} \}

  1. Solve the system of equations:

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

Solutions

  1. The solution to the system of equations is x = 3 and y = 2.

  2. The solution to the system of equations is x = 6 and y = 3.

References

  1. "Linear Algebra and Its Applications" by Gilbert Strang
  2. "Introduction to Linear Algebra" by Jim Hefferon

Glossary

  • System of linear equations: A set of two or more linear equations that are solved simultaneously to find the values of the variables.
  • Linear equation: An equation in which the highest power of the variable is 1.
  • Variable: A value that can change in a mathematical equation.
  • Coefficient: A number that is multiplied by a variable in a mathematical equation.
  • Constant: A number that does not change in a mathematical equation.
    Solving a System of Linear Equations: Q&A =====================================

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of addition. In this article, we will answer some frequently asked questions about solving systems of linear equations.

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 systems of linear equations?

A: There are three main methods for solving systems of linear equations:

  1. Addition method: This method involves adding the two equations to eliminate one of the variables.
  2. Substitution method: This method involves substituting the value of one variable into the other equation to solve for the other variable.
  3. Elimination method: This method involves multiplying one or both of the equations by a constant to eliminate one of the variables.

Q: How do I choose which method to use?

A: The choice of method depends on the specific system of equations. If the coefficients of the variables are the same, the addition method is usually the easiest to use. If the coefficients are different, the substitution or elimination method may be more suitable.

Q: What if I have a system of three or more linear equations?

A: If you have a system of three or more linear equations, you can use the same methods as before, but you may need to use more complex techniques such as substitution or elimination with multiple variables.

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

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

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

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent and cannot be solved simultaneously. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 10.

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

A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and can be solved simultaneously in an infinite number of ways. This can happen if the equations are identical, such as 2x + 3y = 5 and 2x + 3y = 5.

Q: Can I use systems of linear equations to model real-world problems?

A: Yes, systems of linear equations can be used to model a wide range of real-world problems, such as:

  • Finance: Systems of linear equations can be used to model investment portfolios and calculate returns on investment.
  • Science: Systems of linear equations can be used to model population growth and calculate the rate of change of a population.
  • Engineering: Systems of linear equations can be used to model electrical circuits and calculate the current and voltage in a circuit.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have discussed the different methods for solving systems of linear equations, including the addition, substitution, and elimination methods. We have also discussed how to choose which method to use and how to use systems of linear equations to model real-world problems.

Glossary

  • System of linear equations: A set of two or more linear equations that are solved simultaneously to find the values of the variables.
  • Linear equation: An equation in which the highest power of the variable is 1.
  • Variable: A value that can change in a mathematical equation.
  • Coefficient: A number that is multiplied by a variable in a mathematical equation.
  • Constant: A number that does not change in a mathematical equation.
  • Addition method: A method for solving systems of linear equations by adding the two equations to eliminate one of the variables.
  • Substitution method: A method for solving systems of linear equations by substituting the value of one variable into the other equation to solve for the other variable.
  • Elimination method: A method for solving systems of linear equations by multiplying one or both of the equations by a constant to eliminate one of the variables.