Solve The System Of Equations.${ \begin{array}{l} -2x + 15y = -24 \ 2x + 9y = 24 \ x = \square \ y = \square \end{array} }$

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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 substitution and elimination to find the values of the variables.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -2x + 15y = -24 \\ 2x + 9y = 24 \\ x = \square \\ y = \square \end{array} \}

Step 1: 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 2.

{ \begin{array}{l} -2x + 15y = -24 \\ 4x + 18y = 48 \\ x = \square \\ y = \square \end{array} \}

Step 2: Add the Two Equations

Now, let's add the two equations to eliminate the variable x.

{ \begin{array}{l} -2x + 15y = -24 \\ 4x + 18y = 48 \\ 33y = 24 \\ y = \square \end{array} \}

Step 3: Solve for y

Now, let's solve for y by dividing both sides of the equation by 33.

y=2433y=811{ y = \frac{24}{33} \\ y = \frac{8}{11} }

Step 4: Substitute the Value of y into One of the Original Equations

Now, let's substitute the value of y into the first original equation to solve for x.

−2x+15(811)=−24−2x+12011=−24−2x=−24−12011−2x=−264−12011−2x=−38411x=19211{ -2x + 15(\frac{8}{11}) = -24 \\ -2x + \frac{120}{11} = -24 \\ -2x = -24 - \frac{120}{11} \\ -2x = \frac{-264 - 120}{11} \\ -2x = \frac{-384}{11} \\ x = \frac{192}{11} }

Conclusion

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

Real-World Applications

Solving systems of linear equations has many real-world applications, such as:

  • Physics and Engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects, the flow of fluids, and the stress on structures.
  • Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.
  • Economics: Solving systems of linear equations is used to model economic systems, such as supply and demand, and to make predictions about future economic trends.

Tips and Tricks

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

  • Use the method of substitution and elimination: This method is the most efficient way to solve systems of linear equations.
  • Check your work: Always check your work by plugging the values of the variables back into the original equations.
  • Use technology: There are many online tools and software programs that can help you solve systems of linear equations.

Common Mistakes

Here are some common mistakes to avoid when solving systems of linear equations:

  • Not checking your work: Always check your work by plugging the values of the variables back into the original equations.
  • Not using the method of substitution and elimination: This method is the most efficient way to solve systems of linear equations.
  • Not using technology: There are many online tools and software programs that can help you solve systems of linear equations.

Conclusion

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 of solving systems of linear equations?

A: There are two main methods of solving systems of linear equations: the method of substitution and the method of elimination.

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: How do I choose which method to use?

A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.

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

A: Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking your work by plugging the values of the variables back into the original equations.
  • Not using the method of substitution and elimination.
  • Not using technology, such as online tools and software programs, to help you solve systems of linear equations.

Q: How do I check my work when solving systems of linear equations?

A: To check your work, plug the values of the variables back into the original equations and make sure that the equations are true.

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

A: Some real-world applications of solving systems of linear equations include:

  • Physics and Engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects, the flow of fluids, and the stress on structures.
  • Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.
  • Economics: Solving systems of linear equations is used to model economic systems, such as supply and demand, and to make predictions about future economic trends.

Q: How do I use technology to help me solve systems of linear equations?

A: There are many online tools and software programs that can help you solve systems of linear equations, such as:

  • Graphing calculators: These calculators can help you visualize the equations and find the solutions.
  • Online equation solvers: These tools can help you solve systems of linear equations by plugging in the values of the variables.
  • Software programs: These programs can help you solve systems of linear equations by using algorithms and formulas.

Q: What are some tips for solving systems of linear equations?

A: Some tips for solving systems of linear equations include:

  • Use the method of substitution and elimination: This method is the most efficient way to solve systems of linear equations.
  • Check your work: Always check your work by plugging the values of the variables back into the original equations.
  • Use technology: There are many online tools and software programs that can help you solve systems of linear equations.

Conclusion

In conclusion, solving systems of linear equations is an important skill that has many real-world applications. By using the method of substitution and elimination, and by checking your work, you can solve systems of linear equations efficiently and accurately.