Find The Solution \[$(x, Y)\$\] To These Two Equations Using Equality.\[$x - 2y = 4\$\] \[$4x + 3y = 5\$\] Show All Your Work To Get The Answer. Write The Answer Here \[$\rightarrow(, \quad)\$\]

Introduction

In mathematics, solving systems of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore how to find the solution to two linear equations using the method of equality. We will use the given equations: $x - 2y = 4$ and $4x + 3y = 5$ to demonstrate the step-by-step process of solving the system of equations.

Step 1: Write Down the Given Equations

The two linear equations are:

$x - 2y = 4$ ... (Equation 1) $4x + 3y = 5$ ... (Equation 2)

Step 2: Multiply Both Sides of Equation 1 by 4

To eliminate the variable $x$ from Equation 2, we need to multiply both sides of Equation 1 by 4. This will give us:

$4x - 8y = 16$ ... (Equation 3)

Step 3: Add Equation 2 and Equation 3

Now, we will add Equation 2 and Equation 3 to eliminate the variable $x$:

$(4x + 3y) + (4x - 8y) = 5 + 16$ $8x - 5y = 21$ ... (Equation 4)

Step 4: Multiply Both Sides of Equation 1 by 3

To eliminate the variable $y$ from Equation 2, we need to multiply both sides of Equation 1 by 3. This will give us:

$3x - 6y = 12$ ... (Equation 5)

Step 5: Multiply Both Sides of Equation 2 by 2

To eliminate the variable $y$ from Equation 2, we need to multiply both sides of Equation 2 by 2. This will give us:

$8x + 6y = 10$ ... (Equation 6)

Step 6: Add Equation 5 and Equation 6

Now, we will add Equation 5 and Equation 6 to eliminate the variable $y$:

$(3x - 6y) + (8x + 6y) = 12 + 10$ $11x = 22$ ... (Equation 7)

Step 7: Solve for x

Now, we will solve for $x$ by dividing both sides of Equation 7 by 11:

$x = \frac{22}{11}$ $x = 2$ ... (Equation 8)

Step 8: Substitute x into Equation 1

Now, we will substitute $x = 2$ into Equation 1 to solve for $y$:

$2 - 2y = 4$ $-2y = 2$ $y = -1$ ... (Equation 9)

Step 9: Write the Solution

The solution to the system of equations is:

$\rightarrow (2, -1)$

Conclusion

In this article, we have demonstrated how to find the solution to two linear equations using the method of equality. We have used the given equations: $x - 2y = 4$ and $4x + 3y = 5$ to show the step-by-step process of solving the system of equations. The solution to the system of equations is $\rightarrow (2, -1)$.

Introduction

Solving systems of linear equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will address some of the most frequently asked questions (FAQs) about solving systems of linear equations.

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 variables. For example, the system of equations:

$x - 2y = 4$ $4x + 3y = 5$

is a system of two linear equations.

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

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

• Substitution method: This method involves substituting the expression for one variable from one equation into the other equation.
• Elimination method: This method involves eliminating one variable from two equations by adding or subtracting the equations.
• Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
• Equality method: This method involves multiplying both sides of one equation by a constant to make the coefficients of one variable the same in both equations.

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

A: The substitution method involves substituting the expression for one variable from one equation into the other equation, while the elimination method involves eliminating one variable from two equations by adding or subtracting the equations.

Q: How do I choose which method to use?

A: The choice of method depends on the type of equations and the variables involved. If the equations are simple and involve only two variables, the substitution method may be the easiest to use. If the equations involve multiple variables or are more complex, the elimination method may be more effective.

Q: What if I have a system of three or more linear equations?

A: If you have a system of three or more linear equations, you can use the elimination method to eliminate one variable from two equations, and then use the substitution method to solve for the remaining variables.

Q: Can I use a calculator to solve systems of linear equations?

A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "Solve" function on a graphing calculator.

Q: What if I get stuck or make a mistake while solving a system of linear equations?

A: If you get stuck or make a mistake while solving a system of linear equations, don't worry! You can always go back and recheck your work. If you're still having trouble, try using a different method or seeking help from a teacher or tutor.

Q: Are there any real-world applications of solving systems of linear equations?

A: Yes, there are many real-world applications of solving systems of linear equations, such as:

• Physics and engineering: Solving systems of linear equations is used to model and solve problems in physics and engineering, such as motion and force.
• Economics: Solving systems of linear equations is used to model and solve problems in economics, such as supply and demand.
• Computer science: Solving systems of linear equations is used in computer science to solve problems in computer graphics and game development.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the different methods for solving systems of linear equations and how to choose the best method for a given problem, you can become proficient in solving these types of problems and apply them to a wide range of fields.