Solve The System Of Equations:${ \begin{array}{l} y = 2x + 2 \ 3y - 9x = 3 \end{array} }$

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 means 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:

$$\begin{array}{l} y = 2x + 2 \\ 3y - 9x = 3 \end{array}$$

Substitution Method

One way to solve this system of equations is by using the substitution method. 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 first equation for y:

$$y = 2x + 2$$

We can rewrite this equation as:

$$2x + 2 = y$$

Now, let's substitute this expression for y into the second equation:

$$3(2x + 2) - 9x = 3$$

Expanding and simplifying the equation, we get:

$$6x + 6 - 9x = 3$$

Combine like terms:

$$-3x + 6 = 3$$

Subtract 6 from both sides:

$$-3x = -3$$

Divide both sides by -3:

$$x = 1$$

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 first equation:

$$y = 2x + 2$$

Substitute x = 1:

$$y = 2(1) + 2$$

Simplify:

$$y = 4$$

Elimination Method

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

Let's start by multiplying the first equation by 3:

$$3y = 6x + 6$$

Now, let's add this equation to the second equation:

$$3y - 9x = 3$$

$$3y = 6x + 6$$

Add the two equations:

$$6y - 9x = 9x + 6$$

Combine like terms:

$$6y = 15x + 6$$

Now, let's divide both sides by 6:

$$y = \frac{15}{6}x + 1$$

Simplify:

$$y = \frac{5}{2}x + 1$$

Now that we have found the value of y in terms of x, we can substitute x = 1 into this equation to find the value of y:

$$y = \frac{5}{2}(1) + 1$$

Simplify:

$$y = 3$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have found that the solution to the system is x = 1 and y = 4.

Tips and Tricks

• When solving a system of linear equations, it's often helpful to use the substitution method or the elimination method.
• Make sure to check your work by plugging the solution back into the original equations.
• If you're having trouble solving a system of linear equations, try graphing the equations on a coordinate plane to see if they intersect.

Real-World Applications

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

• Physics and Engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
• 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 and make predictions about the behavior of markets.

Common Mistakes

• Not checking work: Make sure to plug the solution back into the original equations to check that it's correct.
• Not using the correct method: Choose the method that's best suited for the problem.
• Not simplifying equations: Make sure to simplify equations as much as possible to make them easier to solve.

Final Thoughts

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 means finding the values of the variables that satisfy all the equations in the system.

Q: What are the two main methods for solving a system of linear equations?

A: The two main methods for solving a system of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

A: The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations in a way that eliminates one of the variables.

Q: How do I choose which method to use?

A: Choose the method that's best suited for the problem. If the equations are easy to solve, the substitution method may be a good choice. If the equations are more complex, the elimination method may be a better option.

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

A: Some common mistakes to avoid include:

• Not checking work: Make sure to plug the solution back into the original equations to check that it's correct.
• Not using the correct method: Choose the method that's best suited for the problem.
• Not simplifying equations: Make sure to simplify equations as much as possible to make them easier to solve.

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 used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
• 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 and make predictions about the behavior of markets.

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 equations are consistent and the variables are not dependent. If the equations are inconsistent or the variables are dependent, the system has no solution.

Q: What is the difference between a dependent and an independent variable?

A: A dependent variable is a variable that is dependent on the other variables in the system. An independent variable is a variable that is not dependent on the other variables in the system.

Q: How do I determine if a variable is dependent or independent?

A: To determine if a variable is dependent or independent, look at the equations and see if the variable is isolated on one side of the equation. If it is, the variable is independent. If it is not, the variable is dependent.

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

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

• Use the substitution method or the elimination method to solve the system.
• Check your work by plugging the solution back into the original equations.
• Simplify equations as much as possible to make them easier to solve.
• Use a graphing calculator or a computer program to help solve the system.

Q: How do I know if a system of linear equations is consistent or inconsistent?

A: A system of linear equations is consistent if the equations have a solution. A system of linear equations is inconsistent if the equations have no solution.

Q: What is the difference between a consistent and an inconsistent system?

A: A consistent system is a system that has a solution. An inconsistent system is a system that has no solution.

Q: How do I determine if a system is consistent or inconsistent?

A: To determine if a system is consistent or inconsistent, look at the equations and see if they have a solution. If they do, the system is consistent. If they do not, the system is inconsistent.