What Is The Solution To The System Of Linear Equations Below?${ \begin{array}{l} x + 4y = 22 \ 2x + Y = 9 \end{array} }$A. { (2,9)$}$ B. { (2,5)$}$ C. { (5,3)$}$ D. { (3,5)$}$

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple linear equations simultaneously. In this article, we will explore a system of two linear equations and provide a step-by-step solution to find the values of the variables.

The System of Linear Equations

The given system of linear equations is:

$$\begin{array}{l} x + 4y = 22 \\ 2x + y = 9 \end{array}$$

Method 1: Substitution Method

One way to solve this system is by using the substitution method. We can solve one equation for one variable and then substitute that expression into the other equation.

Let's solve the second equation for $y$:

$$y = 9 - 2x$$

Now, substitute this expression for $y$ into the first equation:

$$x + 4(9 - 2x) = 22$$

Expand and simplify the equation:

$$x + 36 - 8x = 22$$

Combine like terms:

$$-7x + 36 = 22$$

Subtract 36 from both sides:

$$-7x = -14$$

Divide both sides by -7:

$$x = 2$$

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 second equation:

$$2x + y = 9$$

Substitute $x = 2$:

$$2(2) + y = 9$$

Simplify:

$$4 + y = 9$$

Subtract 4 from both sides:

$$y = 5$$

Method 2: Elimination Method

Another way to solve this system is by using the elimination method. We can multiply both equations by necessary multiples such that the coefficients of $y$'s in both equations are the same:

Multiply the first equation by 1 and the second equation by 4:

$$x + 4y = 22$$

$$8x + 4y = 36$$

Now, subtract the first equation from the second equation:

$$(8x - x) + (4y - 4y) = 36 - 22$$

Simplify:

$$7x = 14$$

Divide both sides by 7:

$$x = 2$$

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:

$$x + 4y = 22$$

Substitute $x = 2$:

$$2 + 4y = 22$$

Subtract 2 from both sides:

$$4y = 20$$

Divide both sides by 4:

$$y = 5$$

Conclusion

In this article, we have solved a system of two linear equations using two different methods: substitution and elimination. We have found that the values of the variables $x$ and $y$ are $x = 2$ and $y = 5$, respectively. This solution satisfies both equations and provides a unique solution to the system.

Answer

The solution to the system of linear equations is $\boxed{(2,5)}$.

Discussion

This system of linear equations can be solved using various methods, including substitution and elimination. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves multiplying both equations by necessary multiples such that the coefficients of $y$'s in both equations are the same, and then subtracting one equation from the other.

Real-World Applications

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

• Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Economics: Systems of linear equations are used to model economic systems, including supply and demand, production, and consumption.
• Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.

Tips and Tricks

When solving systems of linear equations, it's essential to:

• Check for consistency: Ensure that the system has a unique solution, or that the equations are inconsistent.
• Use the correct method: Choose the substitution or elimination method based on the coefficients of the variables.
• Simplify the equations: Simplify the equations by combining like terms and eliminating fractions.

By following these tips and tricks, you can efficiently solve systems of linear equations and apply them to real-world problems.

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 two or more variables. Each equation is a linear equation, which means it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A: To determine the type of solution, you can use the following methods:

• Unique solution: If the system has a unique solution, it means that there is only one set of values for the variables that satisfies both equations. You can use the substitution or elimination method to find the solution.
• No solution: If the system has no solution, it means that there is no set of values for the variables that satisfies both equations. This can happen if the equations are inconsistent, meaning that they contradict each other.
• Infinitely many solutions: If the system has infinitely many solutions, it means that there are an infinite number of sets of values for the variables that satisfy both equations. This can happen if the equations are dependent, meaning that one equation is a multiple of the other.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then subtracting one equation from the other.

Q: How do I choose between the substitution method and the elimination method?

A: To choose between the substitution method and the elimination method, you can look at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, it's easier to use the elimination method. If the coefficients of one variable are different in both equations, it's easier to use the substitution method.

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 consistency: Make sure that the system has a unique solution, or that the equations are inconsistent.
• Not using the correct method: Choose the substitution or elimination method based on the coefficients of the variables.
• Not simplifying the equations: Simplify the equations by combining like terms and eliminating fractions.

Q: How do I apply systems of linear equations to real-world problems?

A: Systems of linear equations can be applied to a wide range of real-world problems, including:

• Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, electrical circuits, and mechanical systems.
• Economics: Systems of linear equations are used to model economic systems, including supply and demand, production, and consumption.
• Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.

Q: What are some advanced topics in systems of linear equations?

A: Some advanced topics in systems of linear equations include:

• Matrix operations: Matrix operations, such as matrix multiplication and inversion, can be used to solve systems of linear equations.
• Determinants: Determinants can be used to find the solution to a system of linear equations.
• Eigenvalues and eigenvectors: Eigenvalues and eigenvectors can be used to solve systems of linear equations.

By understanding these advanced topics, you can solve more complex systems of linear equations and apply them to real-world problems.