Solve The System Of Equations:1. X − 4 Y = − 4 X - 4y = -4 X − 4 Y = − 4 2. 5 X − 4 Y = 12 5x - 4y = 12 5 X − 4 Y = 12 Express Y Y Y In Terms Of X X X : Y = 1 4 X + 1 Y = \frac{1}{4}x + 1 Y = 4 1 ​ X + 1

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 using the method of substitution. We will use the given equations to express one variable in terms of the other, and then use this expression to find the value of the other variable.

The System of Equations

The system of equations we will be solving is:

1. $x - 4y = -4$
2. $5x - 4y = 12$

Step 1: Solve One Equation for One Variable

To solve the system of equations, we can start by solving one equation for one variable. Let's solve the first equation for $x$:

$x - 4y = -4$

Adding $4y$ to both sides of the equation, we get:

$x = 4y - 4$

Step 2: Substitute the Expression into the Second Equation

Now that we have an expression for $x$ in terms of $y$, we can substitute this expression into the second equation:

$5x - 4y = 12$

Substituting $x = 4y - 4$ into the second equation, we get:

$5(4y - 4) - 4y = 12$

Step 3: Simplify the Equation

Expanding the equation, we get:

$20y - 20 - 4y = 12$

Combining like terms, we get:

$16y - 20 = 12$

Adding 20 to both sides of the equation, we get:

$16y = 32$

Step 4: Solve for $y$

Dividing both sides of the equation by 16, we get:

$y = \frac{32}{16}$

Simplifying the fraction, we get:

$y = 2$

Step 5: Express $y$ in Terms of $x$

Now that we have found the value of $y$, we can express $y$ in terms of $x$:

$y = \frac{1}{4}x + 1$

Conclusion

In this article, we solved a system of two linear equations using the method of substitution. We started by solving one equation for one variable, and then substituted this expression into the second equation. We simplified the equation and solved for $y$. Finally, we expressed $y$ in terms of $x$. This is a fundamental concept in mathematics, and solving systems of linear equations is a crucial skill for students and professionals alike.

Real-World Applications

Solving systems of linear equations has many real-world applications. For example, in physics, we can use systems of linear equations to model the motion of objects. In economics, we can use systems of linear equations to model the behavior of markets. In engineering, we can use systems of linear equations to design and optimize systems.

Tips and Tricks

Here are some tips and tricks for solving systems of linear equations:

• Make sure to read the problem carefully and understand what is being asked.
• Choose the method of substitution or elimination wisely.
• Simplify the equation as much as possible before solving for the variable.
• Check your work by plugging the solution back into the original equations.

Common Mistakes

Here are some common mistakes to avoid when solving systems of linear equations:

• Not reading the problem carefully and understanding what is being asked.
• Choosing the wrong method of substitution or elimination.
• Not simplifying the equation enough before solving for the variable.
• Not checking the work by plugging the solution back into the original equations.

Conclusion

Introduction

In our previous article, we discussed how to solve systems of linear equations using the method of substitution. In this article, we will answer some common questions that students and professionals may have when it comes to 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: What are the different methods for solving systems of linear equations?

There are two main methods for solving systems of linear equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

The method of substitution involves solving one equation for one variable and then substituting this expression into the other equation.

Q: What is the method of elimination?

The method of elimination involves adding or subtracting the equations to eliminate one variable and then solving for the other variable.

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

You can choose between the method of substitution and the method of elimination based on the form of the equations. If the equations are in the form of $y = mx + b$, then the method of substitution is usually the best choice. If the equations are in the form of $ax + by = c$, then the method of elimination is usually the best choice.

Q: What are some common mistakes to avoid when solving systems of linear equations?

Some common mistakes to avoid when solving systems of linear equations include:

• Not reading the problem carefully and understanding what is being asked.
• Choosing the wrong method of substitution or elimination.
• Not simplifying the equation enough before solving for the variable.
• Not checking the work by plugging the solution back into the original equations.

Q: How do I check my work when solving systems of linear equations?

To check your work when solving systems of linear equations, you can plug the solution back into the original equations and make sure that it satisfies both equations.

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

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

• Modeling the motion of objects in physics.
• Modeling the behavior of markets in economics.
• Designing and optimizing systems in engineering.

Q: How can I practice solving systems of linear equations?

You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations on your own using real-world data or scenarios.

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

Some advanced topics in solving systems of linear equations include:

• Solving systems of linear equations with more than two variables.
• Solving systems of linear equations with non-linear equations.
• Solving systems of linear equations with complex numbers.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve systems of linear equations using the method of substitution or elimination. Remember to read the problem carefully, choose the right method, simplify the equation, and check your work. With practice and patience, you can become proficient in solving systems of linear equations.

Additional Resources

For more information on solving systems of linear equations, you can check out the following resources:

• Khan Academy: Solving Systems of Linear Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram MathWorld: Systems of Linear Equations

Final Tips

Here are some final tips for solving systems of linear equations:

• Practice, practice, practice! The more you practice, the more comfortable you will become with solving systems of linear equations.
• Don't be afraid to ask for help if you get stuck. There are many resources available online and in textbooks that can help you solve systems of linear equations.
• Remember to check your work by plugging the solution back into the original equations. This will help you ensure that your solution is correct.