Solve The Following System Of Equations:1. $-8x + 4y = -3$2. 8 X − 4 Y = 12 8x - 4y = 12 8 X − 4 Y = 12

Introduction

In mathematics, a system of linear equations is a set of two or more linear 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 method of substitution and elimination to find the solution.

The System of Equations

The system of equations we will be solving is:

1. $-8x + 4y = -3$
2. $8x - 4y = 12$

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:

1. $-8x + 4y = -3$
2. $8x - 4y = 12$

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples. Let's multiply the first equation by 2 and the second equation by 1.

1. $-16x + 8y = -6$
2. $8x - 4y = 12$

Step 3: Add the Equations

Now, let's add the two equations to eliminate the variable x.

$-16x + 8y = -6$ $+ 8x - 4y = 12$

$-8x + 4y = 6$

Step 4: Solve for y

Now that we have eliminated the variable x, we can solve for y. Let's rewrite the equation in terms of y.

$4y = 6$ $y = \frac{6}{4}$ $y = \frac{3}{2}$

Step 5: Substitute the Value of y into One of the Original Equations

Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x. Let's substitute the value of y into the first equation.

$-8x + 4y = -3$ $-8x + 4(\frac{3}{2}) = -3$ $-8x + 6 = -3$ $-8x = -9$ $x = \frac{-9}{-8}$ $x = \frac{9}{8}$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found the values of x and y, which are $x = \frac{9}{8}$ and $y = \frac{3}{2}$, respectively.

Why is Solving Systems of Linear Equations Important?

Solving systems of linear equations is an important skill in mathematics and has many real-world applications. Some of the reasons why solving systems of linear equations is important include:

• Problem-solving: Solving systems of linear equations helps us to solve problems that involve multiple variables and constraints.
• Modeling: Solving systems of linear equations helps us to model real-world situations, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
• Science and Engineering: Solving systems of linear equations is used in many scientific and engineering applications, such as physics, engineering, and computer science.
• Computer Programming: Solving systems of linear equations is used in many computer programming applications, such as game development, computer graphics, and scientific simulations.

Common Applications of Solving Systems of Linear Equations

Some of the common applications of solving systems of linear equations include:

• Physics: Solving systems of linear equations is used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Solving systems of linear equations is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Solving systems of linear equations is used in many computer science applications, such as game development, computer graphics, and scientific simulations.
• Economics: Solving systems of linear equations is used to model economic systems, such as the behavior of supply and demand, and the impact of taxes and subsidies on the economy.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of two linear equations with two variables using the method of substitution and elimination. In this article, we will answer some of the most frequently asked questions 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 are solved simultaneously to find the values of the variables.

Q: Why do we need to solve systems of linear equations?

A: Solving systems of linear equations helps us to solve problems that involve multiple variables and constraints. It is used in many real-world applications, such as physics, engineering, and computer science.

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 value of one variable into the other equation to solve for the other variable.
• Elimination method: This method involves adding or subtracting the equations to eliminate one of the variables.
• Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
• Matrix method: This method involves using matrices to solve the system of equations.

Q: What is the difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3y = 5 is a linear equation. A nonlinear equation is an equation in which the highest power of the variable is greater than 1. For example, x^2 + 3y = 5 is a nonlinear equation.

Q: Can we solve a system of linear equations with more than two variables?

A: Yes, we can solve a system of linear equations with more than two variables. However, it becomes more complex and requires the use of matrices and other advanced techniques.

Q: What is the importance of solving systems of linear equations in real-world applications?

A: Solving systems of linear equations is used in many real-world applications, such as:

• Physics: Solving systems of linear equations is used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Solving systems of linear equations is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Solving systems of linear equations is used in many computer science applications, such as game development, computer graphics, and scientific simulations.
• Economics: Solving systems of linear equations is used to model economic systems, such as the behavior of supply and demand, and the impact of taxes and subsidies on the economy.

Q: Can we use technology to solve systems of linear equations?

A: Yes, we can use technology to solve systems of linear equations. Many calculators and computer software programs, such as MATLAB and Mathematica, have built-in functions for solving systems of linear equations.

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 extraneous solutions: Make sure to check for extraneous solutions, which are solutions that do not satisfy the original equations.
• Not using the correct method: Make sure to use the correct method for solving the system of equations.
• Not checking for consistency: Make sure to check for consistency, which means that the system of equations has a solution.

Conclusion

In conclusion, solving systems of linear equations is an important skill in mathematics and has many real-world applications. We have answered some of the most frequently asked questions about solving systems of linear equations and have discussed the importance of this skill in real-world applications.