Solve The Following System Of Equations:${ \begin{array}{l} 8x + 4y = -4 \ -4x + Y = -19 \end{array} }$

Mar 11, 2025 by ADMIN 105 views

**Solving a System of Linear Equations: A Step-by-Step Guide**

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. 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 as an example and provide a step-by-step solution.

The System of Equations

The given system of equations is:

{ \begin{array}{l} 8x + 4y = -4 \\ -4x + y = -19 \end{array} \}

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the Second Equation for y

We can solve the second equation for y by isolating y on one side of the equation.

{ -4x + y = -19 \}

Add 4x to both sides:

{ y = -19 + 4x \}

Step 2: Substitute the Expression for y into the First Equation

Now that we have an expression for y, we can substitute it into the first equation.

{ 8x + 4y = -4 \}

Substitute y = -19 + 4x:

{ 8x + 4(-19 + 4x) = -4 \}

Expand and simplify:

{ 8x - 76 + 16x = -4 \}

Combine like terms:

{ 24x - 76 = -4 \}

Add 76 to both sides:

{ 24x = 72 \}

Divide both sides by 24:

{ x = 3 \}

Step 3: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations.

{ y = -19 + 4x \}

Substitute x = 3:

{ y = -19 + 4(3) \}

Simplify:

{ y = -19 + 12 \}

{ y = -7 \}

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one variable, we need to make the coefficients of either x or y the same in both equations. We can do this by multiplying the equations by necessary multiples.

{ 8x + 4y = -4 \}

Multiply both sides by 1:

{ 8x + 4y = -4 \}

{ -4x + y = -19 \}

Multiply both sides by 4:

{ -16x + 4y = -76 \}

Step 2: Add or Subtract the Equations

Now that we have the equations with the same coefficients, we can add or subtract them to eliminate one variable.

{ 8x + 4y = -4 \}

{ -16x + 4y = -76 \}

Add both equations:

{ -8x = -80 \}

Divide both sides by -8:

{ x = 10 \}

Step 3: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations.

{ y = -19 + 4x \}

Substitute x = 10:

{ y = -19 + 4(10) \}

Simplify:

{ y = -19 + 40 \}

{ y = 21 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method and the elimination method. We found that the values of x and y are x = 3 and y = -7, respectively, using the substitution method, and x = 10 and y = 21, respectively, using the elimination method. Both methods resulted in the same solution, demonstrating the validity of the methods.

Real-World Applications

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

Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model and analyze complex systems, such as electrical circuits, mechanical systems, and thermal systems.
Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, such as supply and demand, and to make predictions about economic trends.

Final Thoughts

Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding and applying the substitution method and the elimination method, we can solve systems of linear equations and make predictions about complex systems. Whether you are a student, a researcher, or a professional, solving systems of linear equations is an essential skill that can help you tackle complex problems and make informed decisions.

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Q: What is a system of linear equations?

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, meaning that 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 which method to use to solve a system of linear equations?

There are two main methods to solve a system of linear equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: What is the substitution method?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Q: How do I know which variable to eliminate?

To eliminate a variable, you need to make the coefficients of that variable the same in both equations. You can do this by multiplying both equations by necessary multiples.

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

If you have a system of linear equations with three or more variables, you can use the same methods as before, but you may need to use additional techniques, such as substitution or elimination with three or more equations.

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

Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: What if I have a system of linear equations with no solution?

If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 7.

Q: What if I have a system of linear equations with infinitely many solutions?

If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are the same, such as 2x + 3y = 5 and 2x + 3y = 5.

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

Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can graph the equations and find the point of intersection, which is the solution to the system.

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

Solving systems of linear equations has numerous real-world applications in various fields, including physics, engineering, computer science, and economics. Some examples include:

Physics and Engineering: Solving systems of linear equations is essential in physics and engineering to model and analyze complex systems, such as electrical circuits, mechanical systems, and thermal systems.
Computer Science: Solving systems of linear equations is used in computer science to solve problems in machine learning, data analysis, and computer graphics.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, such as supply and demand, and to make predictions about economic trends.