Solve The System Of Equations:$\[ \begin{array}{l} x + 2y = 4 \\ -x - Y = 2 \end{array} \\]

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:

$$\begin{array}{l} x + 2y = 4 \\ -x - y = 2 \end{array}$$

Understanding the System

To solve this system, we need to understand the concept of linear equations and how they can be represented graphically. A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we have two linear equations with two variables, x and y.

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 solve the first equation for x by subtracting 2y from both sides:

$$x = 4 - 2y$$

Step 2: Substitute the Expression for x into the Second Equation

Now, we can substitute the expression for x into the second equation:

$$-(4 - 2y) - y = 2$$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$$-4 + 2y - y = 2$$

Step 4: Combine Like Terms

Combining like terms, we get:

$$-4 + y = 2$$

Step 5: Add 4 to Both Sides

Adding 4 to both sides, we get:

$$y = 6$$

Step 6: Substitute the Value of y into the Expression for x

Now, we can substitute the value of y into the expression for x:

$$x = 4 - 2(6)$$

Step 7: Simplify the Expression

Simplifying the expression, we get:

$$x = 4 - 12$$

Step 8: Combine Like Terms

Combining like terms, we get:

$$x = -8$$

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 Two Equations by Necessary Multiples

We can multiply the two equations by necessary multiples to make the coefficients of y's in both equations the same:

$$\begin{array}{l} x + 2y = 4 \\ 2(-x - y) = 2(-2) \end{array}$$

Step 2: Simplify the Equations

Simplifying the equations, we get:

$$x + 2y = 4$$

$$-2x - 2y = -4$$

Step 3: Add the Two Equations

Adding the two equations, we get:

$$-x = 0$$

Step 4: Solve for x

Solving for x, we get:

$$x = 0$$

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

Now, we can substitute the value of x into one of the original equations:

$$0 + 2y = 4$$

Step 6: Solve for y

Solving for y, we get:

$$y = 2$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: substitution and elimination. We have shown that both methods can be used to solve systems of linear equations, and we have provided a step-by-step guide on how to use each method.

Key Takeaways

• Systems of linear equations are a fundamental concept in mathematics.
• There are two main methods for solving systems of linear equations: substitution and elimination.
• 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.
• Both methods can be used to solve systems of linear equations.

Real-World Applications

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

• Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the flow of fluids.
• Economics: Systems of linear equations are used to model economic systems, such as supply and demand.
• Computer Science: Systems of linear equations are used in computer science, such as in the solution of linear programming problems.

Final Thoughts

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: What are the two main methods for solving systems of linear equations?

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

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations. If the coefficients of one variable are the same in both equations, the elimination method is usually easier to use. If the coefficients of one variable are different in both equations, the substitution method is usually easier to use.

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 same methods as before, but you may need to use additional techniques, such as using matrices or graphing.

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.

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

A: If you get a system of linear equations with no solution, it means that the equations are inconsistent, and there is no value of the variables that satisfies both equations.

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

A: If you get a system of linear equations with infinitely many solutions, it means that the equations are dependent, and there are many values of the variables that satisfy both equations.

Q: Can I use systems of linear equations to model real-world problems?

A: Yes, you can use systems of linear equations to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of economic systems.

Q: What are some common applications of systems of linear equations?

A: Some common applications of systems of linear equations include:

• Physics and Engineering: Systems of linear equations are used to model the motion of objects and the flow of fluids.
• Economics: Systems of linear equations are used to model economic systems, such as supply and demand.
• Computer Science: Systems of linear equations are used in computer science, such as in the solution of linear programming problems.

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

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also use online tools and calculators to help you solve 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 following the order of operations: Make sure to follow the order of operations when solving systems of linear equations.
• Not checking for consistency: Make sure to check for consistency when solving systems of linear equations.
• Not using the correct method: Make sure to use the correct method for solving the system of linear equations.

Conclusion

Solving systems of linear equations is an essential skill for students and professionals alike. By understanding the concept of linear equations and how to solve them, we can apply this knowledge to real-world problems and make informed decisions. In this article, we have provided a list of frequently asked questions and answers about solving systems of linear equations. We hope that this article has been helpful in providing a clear understanding of this important mathematical concept.