Solve The System Of Equations:${ \begin{array}{l} 9x + 6y = -45 \ -12x + 24y = 108 \end{array} }$

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. 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 as an example and provide a step-by-step solution.

The System of Equations

The given system of equations is:

{ \begin{array}{l} 9x + 6y = -45 \\ -12x + 24y = 108 \end{array} \}

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.

Method of Elimination

One of the most common methods for solving a system of linear equations is the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables. In this case, we can multiply the first equation by 4 and the second equation by 3 to make the coefficients of y's in both equations equal.

Step 1: Multiply the Equations

We multiply the first equation by 4 and the second equation by 3 to get:

{ \begin{array}{l} 36x + 24y = -180 \\ -36x + 72y = 324 \end{array} \}

Step 2: Add the Equations

Now, we add the two equations to eliminate the variable x.

{ \begin{array}{l} 36x + 24y = -180 \\ -36x + 72y = 324 \end{array} \}

Adding the two equations, we get:

{ 96y = 144 \}

Step 3: Solve for y

Now, we can solve for y by dividing both sides of the equation by 96.

{ y = \frac{144}{96} \}

Simplifying the equation, we get:

{ y = \frac{3}{2} \}

Step 4: Substitute y into One of the Original Equations

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

{ 9x + 6y = -45 \}

Substituting y = 3/2, we get:

{ 9x + 6(\frac{3}{2}) = -45 \}

Step 5: Solve for x

Now, we can solve for x by simplifying the equation.

{ 9x + 6(\frac{3}{2}) = -45 \}

Simplifying the equation, we get:

{ 9x + 9 = -45 \}

Subtracting 9 from both sides, we get:

{ 9x = -54 \}

Dividing both sides by 9, we get:

{ x = -6 \}

Conclusion

In this article, we solved a system of linear equations using the method of elimination. We multiplied the equations to make the coefficients of y's in both equations equal, added the equations to eliminate the variable x, solved for y, and substituted y into one of the original equations to solve for x. The final solution is x = -6 and y = 3/2.

Applications of Solving Systems of Linear Equations

Solving systems of linear equations has numerous applications in various fields such as physics, engineering, economics, and computer science. Some of the applications include:

  • Physics: Solving systems of linear equations is used to describe the motion of objects in physics. For example, the equations of motion for an object under the influence of gravity can be represented as a system of linear equations.
  • Engineering: Solving systems of linear equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
  • Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of economic variables.
  • Computer Science: Solving systems of linear equations is used in computer graphics, game development, and machine learning.

Real-World Examples

Solving systems of linear equations has numerous real-world applications. Some of the examples include:

  • Traffic Flow: Solving systems of linear equations can be used to model traffic flow and optimize traffic light timing to reduce congestion.
  • Resource Allocation: Solving systems of linear equations can be used to allocate resources such as personnel, equipment, and materials to optimize productivity and efficiency.
  • Financial Planning: Solving systems of linear equations can be used to create financial plans and make predictions about future financial outcomes.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics that has numerous applications in various fields such as physics, engineering, economics, and computer science. The method of elimination is one of the most common methods for solving systems of linear equations. By following the steps outlined in this article, we can solve systems of linear equations and apply the solutions to real-world problems.