Solve The Following System Of Equations:${ \begin{align*} -6x + 6y &= 6 \ -6x + 3y &= -12 \end{align*} }$

Introduction

In mathematics, a system of linear equations is a set of two or more equations in which the variables are linear. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The given system of equations is:

{ \begin{align*} -6x + 6y &= 6 \\ -6x + 3y &= -12 \end{align*} \}

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:

1. $-6x + 6y = 6$
2. $-6x + 3y = -12$

Step 2: Eliminate One Variable

To solve the system of equations, we can eliminate one variable by subtracting one equation from the other. Let's eliminate the variable $x$ by subtracting the second equation from the first equation.

{ \begin{align*} (-6x + 6y) - (-6x + 3y) &= 6 - (-12) \\ 6y - 3y &= 18 \\ 3y &= 18 \end{align*} \}

Step 3: Solve for One Variable

Now that we have eliminated the variable $x$, we can solve for the variable $y$. To do this, we can divide both sides of the equation by 3.

{ \begin{align*} \frac{3y}{3} &= \frac{18}{3} \\ y &= 6 \end{align*} \}

Step 4: Substitute the Value of One Variable

Now that we have found the value of the variable $y$, we can substitute this value into one of the original equations to find the value of the variable $x$. Let's substitute the value of $y$ into the first equation.

{ \begin{align*} -6x + 6(6) &= 6 \\ -6x + 36 &= 6 \\ -6x &= -30 \\ x &= 5 \end{align*} \}

Conclusion

In this article, we have solved a system of two linear equations with two variables. We have used the method of elimination to eliminate one variable and then solved for the other variable. The final solution is $x = 5$ and $y = 6$.

Example Use Cases

Solving a system of linear equations has many practical applications in real-life situations. Here are a few examples:

• Physics and Engineering: Solving a system of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects or the behavior of electrical circuits.
• Economics: Solving a system of linear equations is used in economics to model the behavior of economic systems, such as the supply and demand of goods and services.
• Computer Science: Solving a system of linear equations is used in computer science to solve problems in computer graphics, machine learning, and data analysis.

Tips and Tricks

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

• Use the method of elimination: The method of elimination is a powerful tool for solving a system of linear equations. It involves eliminating one variable by subtracting one equation from the other.
• Use substitution: Substitution is another method for solving a system of linear equations. It involves substituting the value of one variable into one of the original equations to find the value of the other variable.
• Use matrices: Matrices are a powerful tool for solving a system of linear equations. They can be used to represent the coefficients of the equations and the variables.

Conclusion

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations in which the variables are linear. In other words, it is a set of equations in which the variables are raised to the power of 1.

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

A: A system of linear equations has a solution if and only if the two equations are consistent. In other words, if the two equations have the same solution, then the system has a solution.

Q: What is the method of elimination?

A: The method of elimination is a powerful tool for solving a system of linear equations. It involves eliminating one variable by subtracting one equation from the other.

Q: How do I use the method of elimination?

A: To use the method of elimination, follow these steps:

1. Write down the two equations.
2. Subtract one equation from the other to eliminate one variable.
3. Solve for the other variable.
4. Substitute the value of the other variable into one of the original equations to find the value of the first variable.

Q: What is substitution?

A: Substitution is another method for solving a system of linear equations. It involves substituting the value of one variable into one of the original equations to find the value of the other variable.

Q: How do I use substitution?

A: To use substitution, follow these steps:

1. Write down the two equations.
2. Solve for one variable in one of the equations.
3. Substitute the value of the variable into the other equation.
4. Solve for the other variable.

Q: What is a matrix?

A: A matrix is a rectangular array of numbers. It can be used to represent the coefficients of the equations and the variables.

Q: How do I use matrices to solve a system of linear equations?

A: To use matrices to solve a system of linear equations, follow these steps:

1. Write down the two equations.
2. Represent the coefficients of the equations and the variables as a matrix.
3. Use the matrix to solve for the variables.

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

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

• Not checking for consistency: Make sure that the two equations are consistent before solving the system.
• Not using the correct method: Use the correct method for solving the system, such as elimination or substitution.
• Not checking for extraneous solutions: Make sure that the solution is not extraneous by checking it in both equations.

Q: How do I check if a solution is extraneous?

A: To check if a solution is extraneous, substitute the values of the variables into both equations and check if they are true.

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

A: Solving a system of linear equations has many real-world applications, including:

• Physics and Engineering: Solving a system of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects or the behavior of electrical circuits.
• Economics: Solving a system of linear equations is used in economics to model the behavior of economic systems, such as the supply and demand of goods and services.
• Computer Science: Solving a system of linear equations is used in computer science to solve problems in computer graphics, machine learning, and data analysis.

Conclusion

In conclusion, solving a system of linear equations is a fundamental concept in mathematics that has many practical applications in real-life situations. By using the method of elimination, substitution, and matrices, we can solve a system of linear equations and find the values of the variables that satisfy all the equations in the system.