Solve The System Of Equations.${ \begin{array}{l} -8y + 9x = -5 \ 8y + 7x = -75 \ x = \square \ y = \square \end{array} }$

by ADMIN 124 views

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 substitution and elimination to find the values of the variables.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -8y + 9x = -5 \\ 8y + 7x = -75 \\ x = \square \\ y = \square \end{array} \}

Step 1: Write Down the Given Equations

The first step is to write down the given equations.

{ \begin{array}{l} -8y + 9x = -5 \\ 8y + 7x = -75 \end{array} \}

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples.

Let's multiply the first equation by 1 and the second equation by 1.

{ \begin{array}{l} -8y + 9x = -5 \\ 8y + 7x = -75 \end{array} \}

Step 3: Add or Subtract the Equations

Now, we need to add or subtract the equations to eliminate one of the variables.

Let's add the two equations.

{ \begin{array}{l} (-8y + 9x) + (8y + 7x) = -5 + (-75) \\ 16x = -80 \end{array} \}

Step 4: Solve for x

Now, we need to solve for x.

{ \begin{array}{l} 16x = -80 \\ x = \frac{-80}{16} \\ x = -5 \end{array} \}

Step 5: Substitute the Value of x into One of the Original Equations

Now, we need to substitute the value of x into one of the original equations to solve for y.

Let's substitute the value of x into the first equation.

{ \begin{array}{l} -8y + 9(-5) = -5 \\ -8y - 45 = -5 \\ -8y = 40 \\ y = \frac{40}{-8} \\ y = -5 \end{array} \}

Conclusion

In this article, we solved a system of linear equations using the method of substitution and elimination. We first multiplied the equations by necessary multiples, then added or subtracted the equations to eliminate one of the variables. Finally, we solved for the other variable by substituting the value of the first variable into one of the original equations.

Solving a System of Linear Equations: Tips and Tricks

Here are some tips and tricks to help you solve a system of linear equations:

  • Use the method of substitution and elimination: These two methods are the most common methods used to solve a system of linear equations.
  • Multiply the equations by necessary multiples: This will help you eliminate one of the variables.
  • Add or subtract the equations: This will help you eliminate one of the variables.
  • Solve for one variable: Once you have eliminated one of the variables, you can solve for the other variable.
  • Check your answer: Once you have solved for both variables, you can check your answer by plugging the values back into the original equations.

Real-World Applications of Solving a System of Linear Equations

Solving a system of linear equations has many real-world applications. Here are a few examples:

  • Physics: Solving a system of linear equations is used to solve problems in physics, such as finding the position and velocity of an object.
  • Engineering: Solving a system of linear equations is used to solve problems in engineering, such as designing a bridge or a building.
  • Economics: Solving a system of linear equations is used to solve problems in economics, such as finding the equilibrium price and quantity of a good.
  • Computer Science: Solving a system of linear equations is used to solve problems in computer science, such as finding the shortest path in a graph.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of linear equations using the method of substitution and elimination. In this article, we will answer some frequently asked questions about solving a system 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 methods of solving a system of linear equations?

A: There are two main methods of solving a system of linear equations: substitution and elimination.

  • Substitution method: This method involves substituting the value of one variable into one of the original equations to solve for the other variable.
  • Elimination method: This method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I choose which method to use?

A: The choice of method depends on the coefficients of the variables in the equations. If the coefficients are the same, use the substitution method. If the coefficients are different, use the elimination method.

Q: What is the difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A nonlinear equation is an equation in which the highest power of the variable is greater than 1.

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. However, it's always a good idea to check your answer by plugging the values back into the original equations.

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

A: Here are some common mistakes to avoid:

  • Not checking your answer: Always check your answer by plugging the values back into the original equations.
  • Not following the order of operations: Always follow the order of operations (PEMDAS) when solving a system of linear equations.
  • Not using the correct method: Make sure to use the correct method (substitution or elimination) for the given equations.

Q: Can I use a system of linear equations to solve a problem in real life?

A: Yes, you can use a system of linear equations to solve a problem in real life. Here are a few examples:

  • Physics: Solving a system of linear equations is used to solve problems in physics, such as finding the position and velocity of an object.
  • Engineering: Solving a system of linear equations is used to solve problems in engineering, such as designing a bridge or a building.
  • Economics: Solving a system of linear equations is used to solve problems in economics, such as finding the equilibrium price and quantity of a good.

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

A: A system of linear equations has a solution if the equations are consistent and the variables are not dependent. If the equations are inconsistent or the variables are dependent, the system has no solution.

Conclusion

In conclusion, solving a system of linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the methods of substitution and elimination, you can solve a system of linear equations and apply it to real-world problems. With practice and patience, you can become proficient in solving a system of linear equations and answer any questions that may arise.

Additional Resources

Here are some additional resources to help you learn more about solving a system of linear equations:

  • Online tutorials: There are many online tutorials available that can help you learn how to solve a system of linear equations.
  • Practice problems: Practice problems are a great way to reinforce your understanding of solving a system of linear equations.
  • Math textbooks: Math textbooks are a great resource for learning about solving a system of linear equations.

Final Tips

Here are some final tips to help you become proficient in solving a system of linear equations:

  • Practice regularly: Practice regularly to reinforce your understanding of solving a system of linear equations.
  • Use online resources: Use online resources, such as tutorials and practice problems, to help you learn how to solve a system of linear equations.
  • Check your answer: Always check your answer by plugging the values back into the original equations.