Find The Solution For This System Of Equations. Type The Correct Answer In Each Box. If Necessary, Use / For The Fraction Bar(s).$\[ \begin{aligned} 2x - 3y &= 2 \\ x &= 6y - 5 \end{aligned} \\]$\[ \begin{array}{l} x = \\ y =

Introduction

In mathematics, a system of equations is a set of two or more 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 given system of equations to find the values of x and y.

The System of Equations

The given system of equations is:

{ \begin{aligned} 2x - 3y &= 2 \\ x &= 6y - 5 \end{aligned} \}

Step 1: Write Down the Given Equations

We have two equations:

1. $2x - 3y = 2$
2. $x = 6y - 5$

Step 2: Solve One of the Equations for One Variable

We will solve the second equation for x:

$x = 6y - 5$

Step 3: Substitute the Expression for x into the First Equation

Substitute the expression for x into the first equation:

$2(6y - 5) - 3y = 2$

Step 4: Simplify the Equation

Expand and simplify the equation:

$12y - 10 - 3y = 2$

Combine like terms:

$9y - 10 = 2$

Step 5: Add 10 to Both Sides of the Equation

Add 10 to both sides of the equation:

$9y = 12$

Step 6: Divide Both Sides of the Equation by 9

Divide both sides of the equation by 9:

$y = \frac{12}{9}$

Simplify the fraction:

$y = \frac{4}{3}$

Step 7: Substitute the Value of y into One of the Original Equations

Substitute the value of y into the second original equation:

$x = 6(\frac{4}{3}) - 5$

Step 8: Simplify the Equation

Simplify the equation:

$x = 8 - 5$

$x = 3$

Conclusion

We have solved the system of equations and found the values of x and y:

$x = 3$ $y = \frac{4}{3}$

Discussion

Solving a system of equations is an essential skill in mathematics and is used in various fields such as physics, engineering, and economics. In this article, we have used a step-by-step approach to solve a system of two linear equations with two variables. We have substituted the expression for x into the first equation, simplified the equation, and solved for y. Finally, we have substituted the value of y into one of the original equations to find the value of x.

Tips and Tricks

• When solving a system of equations, it is essential to check the solution by substituting the values of x and y into both original equations.
• If the system of equations has no solution, it means that the equations are inconsistent.
• If the system of equations has infinitely many solutions, it means that the equations are dependent.

Real-World Applications

Solving a system of equations has numerous real-world applications, such as:

• Physics: Solving a system of equations is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Solving a system of equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
• Economics: Solving a system of equations is used to model economic systems and make predictions about the behavior of economic variables.

Conclusion

Introduction

In our previous article, we solved a system of two linear equations with two variables. In this article, we will answer some frequently asked questions about solving a system of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

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

A system of equations has a solution if the equations are consistent and have the same solution. If the equations are inconsistent, the system has no solution. If the equations are dependent, the system has infinitely many solutions.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A system of linear equations is a set of equations where each equation is a linear equation. A system of nonlinear equations is a set of equations where each equation is a nonlinear equation.

Q: How do I solve a system of nonlinear equations?

Solving a system of nonlinear equations is more complex than solving a system of linear equations. There are several methods to solve a system of nonlinear equations, including:

• Substitution method
• Elimination method
• Graphical method
• Numerical method

Q: What is the substitution method?

The substitution method is a method of solving a system of equations by substituting the expression for one variable into the other equation.

Q: What is the elimination method?

The elimination method is a method of solving a system of equations by eliminating one variable by adding or subtracting the equations.

Q: What is the graphical method?

The graphical method is a method of solving a system of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the numerical method?

The numerical method is a method of solving a system of equations by using numerical methods such as the Newton-Raphson method.

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

To check if a solution is correct, substitute the values of the variables into both original equations and check if the equations are true.

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

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

• Not checking if the equations are consistent
• Not checking if the solution is correct
• Not using the correct method to solve the system
• Not simplifying the equations before solving

Q: How do I use a system of equations in real-world applications?

A system of equations is used in various real-world applications, including:

• Physics: Solving a system of equations is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Solving a system of equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
• Economics: Solving a system of equations is used to model economic systems and make predictions about the behavior of economic variables.

Conclusion

In conclusion, solving a system of equations is an essential skill in mathematics that has numerous real-world applications. By understanding the different methods of solving a system of equations and avoiding common mistakes, we can solve systems of equations and make predictions about the behavior of economic variables.

Tips and Tricks

• Always check if the equations are consistent before solving
• Always check if the solution is correct
• Use the correct method to solve the system
• Simplify the equations before solving
• Use a system of equations to model real-world applications

Real-World Applications

Solving a system of equations has numerous real-world applications, including:

• Physics: Solving a system of equations is used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Solving a system of equations is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
• Economics: Solving a system of equations is used to model economic systems and make predictions about the behavior of economic variables.

Conclusion

In conclusion, solving a system of equations is an essential skill in mathematics that has numerous real-world applications. By understanding the different methods of solving a system of equations and avoiding common mistakes, we can solve systems of equations and make predictions about the behavior of economic variables.