Solve The System Of Equations:$\[ \begin{cases} 4x + 2y = -12 \\ 3x + Y = -10 \end{cases} \\]

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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 solution to the system.

The System of Equations

The system of equations we will be solving is:

{ \begin{cases} 4x + 2y = -12 \\ 3x + y = -10 \end{cases} \}

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 Substitution

One way to solve this system is by using the method of substitution. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Let's start by solving the second equation for y:

3x+y=−103x + y = -10

Subtracting 3x from both sides gives:

y=−10−3xy = -10 - 3x

Now, substitute this expression for y into the first equation:

4x+2y=−124x + 2y = -12

Substituting y=−10−3xy = -10 - 3x into the first equation gives:

4x+2(−10−3x)=−124x + 2(-10 - 3x) = -12

Expanding and simplifying the equation gives:

4x−20−6x=−124x - 20 - 6x = -12

Combine like terms:

−2x−20=−12-2x - 20 = -12

Adding 20 to both sides gives:

−2x=8-2x = 8

Dividing both sides by -2 gives:

x=−4x = -4

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the second equation:

3x+y=−103x + y = -10

Substituting x=−4x = -4 into the second equation gives:

3(−4)+y=−103(-4) + y = -10

Expanding and simplifying the equation gives:

−12+y=−10-12 + y = -10

Adding 12 to both sides gives:

y=2y = 2

Method of Elimination

Another way to solve this system is by using the method of elimination. This method involves adding or subtracting the equations to eliminate one of the variables.

Let's start by multiplying the second equation by 2 to make the coefficients of y in both equations the same:

2(3x+y)=2(−10)2(3x + y) = 2(-10)

Expanding and simplifying the equation gives:

6x+2y=−206x + 2y = -20

Now, add the two equations together to eliminate y:

(4x+2y)+(6x+2y)=−12+(−20)(4x + 2y) + (6x + 2y) = -12 + (-20)

Combine like terms:

10x+4y=−3210x + 4y = -32

Subtracting 4y from both sides gives:

10x=−32−4y10x = -32 - 4y

Dividing both sides by 10 gives:

x=−32−4y10x = \frac{-32 - 4y}{10}

Now, substitute this expression for x into one of the original equations to find the value of y. Let's use the first equation:

4x+2y=−124x + 2y = -12

Substituting x=−32−4y10x = \frac{-32 - 4y}{10} into the first equation gives:

4(−32−4y10)+2y=−124\left(\frac{-32 - 4y}{10}\right) + 2y = -12

Expanding and simplifying the equation gives:

−128−16y10+2y=−12\frac{-128 - 16y}{10} + 2y = -12

Multiplying both sides by 10 to eliminate the fraction gives:

−128−16y+20y=−120-128 - 16y + 20y = -120

Combine like terms:

4y=84y = 8

Dividing both sides by 4 gives:

y=2y = 2

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found that the values of x and y that satisfy both equations are x = -4 and y = 2.

Final Answer

The final answer is x = -4 and y = 2.

Solved System of Equations

The solved system of equations is:

{ \begin{cases} 4x + 2y = -12 \\ 3x + y = -10 \end{cases} \}

x = -4 and y = 2.

Key Takeaways

  • 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.
  • The method of substitution involves solving one of the equations for one variable and then substituting that expression into the other equation.
  • The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
  • The values of x and y that satisfy both equations are x = -4 and y = 2.

Real-World Applications

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

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
  • Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
  • Computer Science: Systems of linear equations are used in computer graphics and game development to create realistic simulations.

Tips and Tricks

  • When solving systems of linear equations, it's often helpful to use the method of substitution or elimination.
  • Make sure to check your work by plugging the values of x and y back into the original equations.
  • Systems of linear equations can be solved using a variety of methods, including graphing, substitution, and elimination.

Conclusion

In conclusion, solving systems of linear equations is an important skill in mathematics and has many real-world applications. By using the method of substitution or elimination, we can find the values of x and y that satisfy both equations. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to a variety of real-world problems.

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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: How do I know which method to use to solve a system of linear equations?

A: The choice of method depends on the specific system of equations. If the coefficients of one variable are the same in both equations, the method of elimination is often the easiest to use. If the coefficients of one variable are different in both equations, the method of substitution is often the easiest to use.

Q: What is the difference between the method of substitution and the method of elimination?

A: The method of substitution involves solving one of the equations for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

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 lines represented by the equations intersect at a single point. If the lines are parallel, the system has no solution.

Q: What is the significance of the term "linear" in the context of linear equations?

A: The term "linear" refers to the fact that the equations are in the form of a straight line. Linear equations have a constant slope and a constant y-intercept.

Q: Can a system of linear equations have more than one solution?

A: No, a system of linear equations can only have one solution. If the lines represented by the equations intersect at a single point, that point is the solution to the system.

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

A: To check your work, plug the values of x and y back into the original equations. If the equations are true, then your solution is correct.

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 and engineering, economics, and computer science.

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

A: Some common mistakes to avoid include:

  • Not checking your work
  • Not using the correct method for the specific system of equations
  • Not following the order of operations
  • Not simplifying the equations before solving

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own using a calculator or a computer program.

Q: What are some advanced topics related to solving systems of linear equations?

A: Some advanced topics related to solving systems of linear equations include:

  • Solving systems of nonlinear equations
  • Solving systems of equations with more than two variables
  • Using matrices to solve systems of linear equations
  • Using computer programs to solve systems of linear equations

Conclusion

In conclusion, solving systems of linear equations is an important skill in mathematics and has many real-world applications. By understanding the different methods for solving systems of linear equations and practicing regularly, you can become proficient in solving these types of problems.