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Introduction to the System of Equations

A system of equations is a set of two or more equations that contain the same variables. In this case, we have a system of two linear equations with two variables, $x$ and $y$. The first equation is $3x - y = 22$, and the second equation is $y = x - 14$. Our goal is to find the value of $y$ that satisfies both equations.

Understanding the Equations

The first equation, $3x - y = 22$, is a linear equation in two variables. It can be rewritten as $y = 3x - 22$. This equation represents a line in the coordinate plane, where the slope is $3$ and the $y$-intercept is $-22$.

The second equation, $y = x - 14$, is also a linear equation in two variables. It represents a line in the coordinate plane, where the slope is $1$ and the $y$-intercept is $-14$.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution. We can substitute the expression for $y$ from the second equation into the first equation. This will give us an equation with only one variable, $x$.

Substituting $y = x - 14$ into the first equation, we get:

$3x - (x - 14) = 22$

Simplifying the equation, we get:

$3x - x + 14 = 22$

Combine like terms:

$2x + 14 = 22$

Subtract $14$ from both sides:

$2x = 8$

Divide both sides by $2$:

$x = 4$

Finding the Value of $y$

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. We will use the second equation, $y = x - 14$.

Substituting $x = 4$ into the equation, we get:

$y = 4 - 14$

Simplifying the equation, we get:

$y = -10$

Conclusion

In this article, we have solved a system of two linear equations with two variables, $x$ and $y$. We used the method of substitution to find the value of $x$, and then substituted it into one of the original equations to find the value of $y$. The value of $y$ is $-10$.

Final Answer

The final answer is: $\boxed{-10}$

Introduction

Solving systems of equations can be a challenging task, especially for those who are new to algebra. However, with the right approach and techniques, it can be a breeze. In this article, we will answer some of the most frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables. In this case, we have a system of two linear equations with two variables, $x$ and $y$.

Q: What are the different methods for solving systems of equations?

A: There are several methods for solving systems of equations, including:

• Substitution method: This method involves substituting the expression for one variable into the other equation.
• Elimination method: This method involves adding or subtracting the two equations to eliminate one of the variables.
• Graphical method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable into the other equation. For example, if we have the equations $y = x - 14$ and $3x - y = 22$, we can substitute the expression for $y$ from the first equation into the second equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the two equations to eliminate one of the variables. For example, if we have the equations $y = x - 14$ and $3x - y = 22$, we can add the two equations to eliminate the variable $y$.

Q: What is the graphical method?

A: The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection. This method is useful for solving systems of linear equations.

Q: How do I choose the right method for solving a system of equations?

A: The choice of method depends on the type of equations and the variables involved. If the equations are linear and have two variables, the substitution or elimination method may be the best choice. If the equations are non-linear or have more than two variables, the graphical method may be more suitable.

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

A: Some common mistakes to avoid when solving systems of equations include:

• Not checking the solution: Make sure to check the solution by substituting it back into both equations.
• Not using the correct method: Choose the right method for the type of equations and variables involved.
• Not simplifying the equations: Simplify the equations before solving them.

Q: How do I check the solution?

A: To check the solution, substitute the values of $x$ and $y$ back into both equations. If the solution satisfies both equations, it is the correct solution.

Q: What are some real-world applications of solving systems of equations?

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

• Physics and engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
• Economics: Solving systems of equations is used to model economic systems and make predictions about the behavior of markets.
• Computer science: Solving systems of equations is used in computer science to solve problems in computer graphics, game development, and machine learning.

Final Answer

The final answer is: Solving systems of equations is a powerful tool for modeling and solving real-world problems.