Solve The System Of Equations:$\[ \begin{cases} -4x - 2y = -12 \\ 4x + 8y = -24 \end{cases} \\]

Feb 28, 2025 by ADMIN 96 views

**Solving Systems of Linear Equations: A Step-by-Step Guide**

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. 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 guide on how to solve it.

The System of Equations

The given system of equations is:

{ \begin{cases} -4x - 2y = -12 \\ 4x + 8y = -24 \end{cases} \}

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.

Method 1: Substitution Method

One way to solve this system 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 first equation for x

We can start by solving the first equation for x:

{ -4x - 2y = -12 \}

Add 2y to both sides:

{ -4x = -12 + 2y \}

Divide both sides by -4:

{ x = \frac{-12 + 2y}{-4} \}

Simplify the expression:

{ x = 3 - \frac{1}{2}y \}

Step 2: Substitute the expression for x into the second equation

Now that we have an expression for x, we can substitute it into the second equation:

{ 4x + 8y = -24 \}

Substitute x = 3 - (1/2)y:

{ 4(3 - \frac{1}{2}y) + 8y = -24 \}

Expand and simplify:

{ 12 - 2y + 8y = -24 \}

Combine like terms:

{ 6y = -36 \}

Divide both sides by 6:

{ y = -6 \}

Step 3: Find the value of x

Now that we have the value of y, we can find the value of x by substituting y into the expression we found in Step 1:

{ x = 3 - \frac{1}{2}(-6) \}

Simplify the expression:

{ x = 3 + 3 \}

{ x = 6 \}

Method 2: Elimination Method

Another way to solve this system 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:

{ \begin{cases} -4x - 2y = -12 \\ 4x + 8y = -24 \end{cases} \}

Multiply the first equation by 2 and the second equation by 1:

{ \begin{cases} -8x - 4y = -24 \\ 4x + 8y = -24 \end{cases} \}

Step 2: Add the equations

Now that we have the equations with the same coefficients for y, we can add them to eliminate y:

{ -8x - 4y + 4x + 8y = -24 + (-24) \}

Simplify the equation:

{ -4x + 4y = -48 \}

Step 3: Solve for x

Now that we have an equation with only one variable, we can solve for x:

{ -4x = -48 - 4y \}

Divide both sides by -4:

{ x = \frac{-48 - 4y}{-4} \}

Simplify the expression:

{ x = 12 + y \}

Step 4: Find the value of y

We can find the value of y by substituting x into one of the original equations. Let's use the first equation:

{ -4x - 2y = -12 \}

Substitute x = 12 + y:

{ -4(12 + y) - 2y = -12 \}

Expand and simplify:

{ -48 - 4y - 2y = -12 \}

Combine like terms:

{ -6y = 36 \}

Divide both sides by -6:

{ y = -6 \}

Step 5: Find the value of x

Now that we have the value of y, we can find the value of x by substituting y into the expression we found in Step 3:

{ x = 12 + (-6) \}

Simplify the expression:

{ x = 6 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We found that the values of x and y that satisfy both equations are x = 6 and y = -6. These values can be verified by substituting them into both equations.

Tips and Tricks

When solving systems of linear equations, it's essential to check your work by substituting the values of x and y into both equations.
The substitution method is often easier to use when one equation is already solved for one variable.
The elimination method is often easier to use when the coefficients of either x or y are the same in both equations.
When using the elimination method, make sure to multiply the equations by necessary multiples to eliminate one variable.

Real-World Applications

Systems of linear equations have numerous real-world applications, including:

Physics: Systems of linear equations are used to describe the motion of objects in two or three dimensions.
Engineering: Systems of linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Systems of linear equations are used to model economic systems and make predictions about future trends.

Final Thoughts

Introduction

In our previous article, we discussed how to solve systems of linear equations using the substitution method and the elimination method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

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: How do I know which method to use?

The choice of method depends on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, the elimination method is often easier to use. If one equation is already solved for one variable, the substitution method is often easier to use.

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

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

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

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: How do I check my work?

To check your work, substitute the values of the variables into both equations and make sure that the equations are true. If the equations are not true, you may need to recheck your work or try a different method.

Q: What if I get stuck?

If you get stuck, try breaking down the problem into smaller steps or using a different method. You can also ask a teacher or tutor for help.

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

Yes, there are many real-world applications of solving systems of linear equations, including:

Physics: Systems of linear equations are used to describe the motion of objects in two or three dimensions.
Engineering: Systems of linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Systems of linear equations are used to model economic systems and make predictions about future trends.

Q: Can I use systems of linear equations to solve problems in other areas of mathematics?

Yes, you can use systems of linear equations to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Q: Are there any tips or tricks for solving systems of linear equations?

Yes, here are a few tips and tricks for solving systems of linear equations:

Make sure to check your work by substituting the values of the variables into both equations.
Use the substitution method when one equation is already solved for one variable.
Use the elimination method when the coefficients of one variable are the same in both equations.
Break down the problem into smaller steps if you get stuck.
Ask a teacher or tutor for help if you need it.

Conclusion

Solving systems of linear equations is a fundamental skill that is used in many areas of mathematics and science. By using the substitution method and the elimination method, we can solve systems of linear equations and find the values of the variables that satisfy both equations. With practice and patience, you can become proficient in solving systems of linear equations and apply this skill to real-world problems.

Additional Resources

If you need additional help or resources, here are a few suggestions:

Khan Academy: Khan Academy has a comprehensive video series on solving systems of linear equations.
Mathway: Mathway is an online calculator that can help you solve systems of linear equations.
Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve systems of linear equations and other mathematical problems.