Solve The Following System Of Equations:${ \begin{cases} 6x - 4y = 18 \ -x - 6y = 7 \end{cases} }$
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. 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 system of equations we will be solving is:
{ \begin{cases} 6x - 4y = 18 \\ -x - 6y = 7 \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 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 in the system to eliminate one of the variables.
To apply the method of elimination, we need to multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same:
{ \begin{cases} 6x - 4y = 18 \\ 3x + 18y = 21 \end{cases} \}
Now, we can add the two equations to eliminate the variable y:
{ (6x - 4y) + (3x + 18y) = 18 + 21 \}
Simplifying the equation, we get:
{ 9x + 14y = 39 \}
However, we are not done yet. We need to eliminate the variable y from the original equations. To do this, we can multiply the first equation by 3 and the second equation by 2:
{ 18x - 12y = 54 \\ -2x - 12y = 14 \}
Now, we can add the two equations to eliminate the variable y:
{ (18x - 12y) + (-2x - 12y) = 54 + 14 \}
Simplifying the equation, we get:
{ 16x = 68 \}
Solving for x
Now that we have eliminated the variable y, we can solve for x. To do this, we can divide both sides of the equation by 16:
{ x = \frac{68}{16} \}
Simplifying the fraction, we get:
{ x = \frac{17}{4} \}
Solving for y
Now that we have found the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:
{ 6x - 4y = 18 \}
Substituting x = 17/4, we get:
{ 6(\frac{17}{4}) - 4y = 18 \}
Simplifying the equation, we get:
{ \frac{102}{4} - 4y = 18 \}
Multiplying both sides of the equation by 4, we get:
{ 102 - 16y = 72 \}
Subtracting 102 from both sides of the equation, we get:
{ -16y = -30 \}
Dividing both sides of the equation by -16, we get:
{ y = \frac{30}{16} \}
Simplifying the fraction, we get:
{ y = \frac{15}{8} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of elimination. We have found the values of x and y that satisfy both equations, which are x = 17/4 and y = 15/8.
Final Answer
The final answer is:
{ \begin{cases} x = \frac{17}{4} \\ y = \frac{15}{8} \end{cases} \}
Example Use Cases
Solving a system of linear equations has many practical applications in various fields, including:
- Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
- Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
- Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand, and to make predictions about economic trends.
Tips and Tricks
Here are some tips and tricks for solving systems of linear equations:
- Use the method of elimination: The method of elimination is a powerful tool for solving systems of linear equations. It involves adding or subtracting the equations in the system to eliminate one of the variables.
- Use substitution: Substitution is another method for solving systems of linear equations. It involves substituting one of the variables in one of the equations with an expression from another equation.
- Use matrices: Matrices are a powerful tool for solving systems of linear equations. They can be used to represent the coefficients of the equations and to perform operations on the equations.
Conclusion
Solving a system of linear equations is a fundamental concept in mathematics that has many practical applications in various fields. By using the method of elimination, substitution, and matrices, we can solve systems of linear equations and find the values of the variables that satisfy all the equations in the system.
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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 involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.
Q: What are the different methods for solving systems of linear equations?
A: There are several methods for solving systems of linear equations, including:
- Method of elimination: This method involves adding or subtracting the equations in the system to eliminate one of the variables.
- Method of substitution: This method involves substituting one of the variables in one of the equations with an expression from another equation.
- Method of matrices: This method involves using matrices to represent the coefficients of the equations and to perform operations on the equations.
Q: How do I choose the method of elimination?
A: To choose the method of elimination, you need to look at the coefficients of the variables in the equations and decide which variable to eliminate first. You can eliminate the variable with the smallest coefficient or the variable that appears most frequently in the equations.
Q: How do I use the method of substitution?
A: To use the method of substitution, you need to choose one of the equations and substitute one of the variables in that equation with an expression from another equation. You can then solve for the other variable and substitute it back into the original equation to find the value of the first variable.
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 for extraneous solutions: Make sure to check for extraneous solutions by plugging the values of the variables back into the original equations.
- Not using the correct method: Make sure to use the correct method for solving the system of linear equations.
- Not simplifying the equations: Make sure to simplify the equations before solving them.
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, you need to plug the values of the variables back into the original equations and make sure that they satisfy all the equations in the system.
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: Solving systems of linear equations is essential in physics and engineering to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
- Computer science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
- Economics: Solving systems of linear equations is used in economics to model economic systems, such as supply and demand, and to make predictions about economic trends.
Q: How do I use matrices to solve systems of linear equations?
A: To use matrices to solve systems of linear equations, you need to represent the coefficients of the equations as a matrix and then perform operations on the matrix to find the values of the variables.
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 elimination: The method of elimination is a powerful tool for solving systems of linear equations.
- Use substitution: Substitution is another method for solving systems of linear equations.
- Use matrices: Matrices are a powerful tool for solving systems of linear equations.
Q: How do I know which method to use?
A: To know which method to use, you need to look at the coefficients of the variables in the equations and decide which method is most suitable for the problem.
Q: What are some common pitfalls to avoid when solving systems of linear equations?
A: Some common pitfalls to avoid when solving systems of linear equations include:
- Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the values of the variables back into the original equations.
- Not using the correct method: Make sure to use the correct method for solving the system of linear equations.
- Not simplifying the equations: Make sure to simplify the equations before solving them.
Q: How do I verify my solution?
A: To verify your solution, you need to plug the values of the variables back into the original equations and make sure that they satisfy all the equations in the system.
Q: What are some resources for learning more about solving systems of linear equations?
A: Some resources for learning more about solving systems of linear equations include:
- Textbooks: There are many textbooks available that cover the topic of solving systems of linear equations.
- Online resources: There are many online resources available that provide tutorials and examples on solving systems of linear equations.
- Practice problems: Practice problems are a great way to learn and reinforce your understanding of solving systems of linear equations.